WEBVTT mathematics/probability/murray
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Hi and welcome back to www.educator.com, these are the probability lectures - my name is Will Murray.
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Today, we are going to talk about making choices and that is going to lead us into combinations and permutations.
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I want to jump right in here.
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There are lots of problems in probability where they say something
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like how many different ways are there to choose from?
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These problems are some of the most confusing ones in probability.
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The reason is that the wording is very subtle and there are two very important distinctions
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that you have to ask about every one of these choosing questions.
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Let me try to walk you through that and in particular,
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I want to draw attention to these two very subtle distinctions.
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Sometimes it is very hard to tell from the wording of the problem but it makes a big difference to the answers.
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Those two subtle distinctions are, are you choosing with the replacement or without replacement.
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Are you making an ordered choice or an unordered choice?
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I want to explain those, explain the differences between those and
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give you some examples of each one so that you can understand what the difference is.
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When you get probability questions, you can make sure that you are understanding the question
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and that you are answering the right question with the right formula.
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I will explain all those differences and then give you all the formulas
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that you need to answer any combination of these questions.
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And then we will work through some examples and hopefully,
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it will all start to be a bit clearer to you by the end of the lecture.
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Our first question here is when you are choosing several things,
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are you choosing with replacement or without replacement?
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The question here is can you choose the same thing more than once?
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If you can choose the same thing more than once, you are choosing with replacement.
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That means after you choose something, that choice goes back into the pool and you can choose it again if you like.
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It gets replaced in the pool and you can choose it again if you like.
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I have couple of examples here to show the distinction between those.
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Suppose you are buying bagels in a bakery and you are choosing do I want an onion bagel?
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Do I want a blueberry bagel? Do I want to a poppy seed bagel?
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And you are going to buy a bag of bagels and take them back to your friends.
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You will say, first, I’m going to buy onion bagel.
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Now, can you choose another onion bagel?
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Sure, because the bakery has a whole shelf of onion bagels.
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That is choosing with the replacement because after you choose the onion bagel,
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you can still choose more onion bagels if you want to.
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Here is an example of choosing without replacement.
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You have a bunch of athletes on the side of a basketball court and
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you want to choose 5 people to be your basketball team.
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You pick the first person and then you want to go back and choose some more,
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can you pick the same person again?
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No, because that person is already on your team.
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That person does not get replaced in the pool after you have chosen that first person.
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That is choosing without replacement.
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We will have different formulas based on whether you are choosing with replacement or without replacement.
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That affects the answer, whether you can make the same choice more than once.
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That is the first distinction.
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Whenever you get a choosing problem, you have to say am I choosing with replacement or without replacement?
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You have to understand that before you can even start to calculate the answer.
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That is the first thing you want to ask, with or without replacement.
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The second thing you want to ask is ordered or unordered?
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Let us talk about that one next.
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And then after that, we will get into some formulas and some actual examples.
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We will calculate some actual problems.
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The second decision you have to make based on the wording of the problem is, is this an ordered or unordered selection?
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In other words, if I choose this thing and then that thing,
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should that be considered different from choosing that thing first and then this thing?
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That is very subtle, it is often not obvious from the wording of the problem.
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But if you are counting those differently then it is an ordered choice.
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If you are counting those to be the same, it is considered an unordered choice.
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Let me give you an example of that.
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It is 2 very similar examples but it will show the subtle distinction.
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Let us suppose we are picking a basketball team and we got 20 people sending on the sidelines
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and we want to pick our basketball team.
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A basketball team is 5 players.
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You have to put 5 people on the court for a basketball team.
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Let us suppose, first of all that this is just a casual, friendly, pickup basketball game in the park.
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We just go out to the park, there are no formal formations.
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We are just going to get 5 people, they are going to run onto the court.
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They are going to throw the basketball back and forth and somebody is going to shoot the ball for the basket.
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That means we are going to pick 5 people and we are going to throw them onto the court, just sort of randomly.
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That is an unordered choice because if I pick Tom, and then Dick, and then Harry to be my basketball team,
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or if I pick Harry and then Tom and then Dick to be my basketball team, it is still the same 3 guys running on the court.
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They are still going to run up and down the court.
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It is the same basketball team either way.
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If I pick 5 guys in one order.
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If I pick 5 guys in a different order, it is still the same basketball team
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because we are not being very formal about this.
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By contrast, suppose we are going to play a formal game, this is a regulation game
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where everybody got positions and everybody is going to stick to their roles.
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You may not be an expert on basketball but there are 5 positions in basketball.
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The positions are there is a center, the guy who stands under the basket.
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There is a power forward, there is a small forward.
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There is a point guard and that should have said shooting guard right there.
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Let me just fix that because that is different from the point guard.
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That says shooting guard.
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There are 5 different positions and you are going to pick 5 players to be your basketball team.
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It matters who plays which position.
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This is different from the informal game on the park.
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You can pick Tom to be your center and then you can pick Harry to be your point guard,
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and then you can fill in the other 3 positions.
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If you pick those same people but in a different order.
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If you pick Harry first, that means Harry.
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You just picked Harry to be your center and then you picked Tom to be your point guard.
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In a formal game, that is a different team because putting Harry in center and
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Tom as point guard is a different team from putting Tom at center and Harry at point guard.
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Even though you got the same guys on the court.
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That is an ordered choice, you are counting those to be different configurations.
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That is what it means to make an ordered choice.
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From now on, whenever you get a probability problem and it has to do with choosing things,
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you got to ask does the order make a difference?
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Does it matter if I pick Tom first and Harry second, and then that means I got one team
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with Tom at center and Harry at point guard, that is an ordered choice.
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If we are just playing an unorganized game in the park, and we are just going to throw Tom and Harry on the court,
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it does not matter who goes on there first, that is an ordered choice.
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And we are going to have a different set of formulas for ordered choices.
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Another kind of common example of this is when you are drawing cards for a poker hand.
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If you are playing poker and you get 5 cards out of the deck, you get your poker hand
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and the question is that an unordered choice or an ordered choice.
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And the answer is that is an unordered because for a poker hand,
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you just get 5 cards in your hand and you can shuffle them around after that, if you like.
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The order that you draw the cards does not matter.
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It is an unordered choice.
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We are going to use those two key decisions to get some formulas
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and see where they lead us and do some examples.
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Let us talk about combinations.
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We are going to learn out combinations and permutations.
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These combinations to count the number of ways to choose a group of unordered objects from N possibilities.
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That means we got N possibilities out there and we are going to choose R,
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make R choices from those N possibilities.
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The important thing here is that we are doing this without replacement.
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Once we choose something, it does not go back into the pool.
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You cannot choose the same thing again.
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And we have a formula for the number of combinations.
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We saw this back in one of the previous lectures here on probability.
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I have showed you where this formula comes from.
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There are two different notations that are very commonly used.
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This is the binomial coefficient notation.
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This is known as a binomial coefficient.
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It is called that because if you expand out the binomial theorem, you get these numbers.
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This is the expression that appears as the coefficients in the binomial theorem.
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This is called the binomial coefficient.
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It is read as N choose R.
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When you read this, you say N choose R.
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That terminology reflects the fact that it comes from choosing R things from N possibilities.
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The other notation for the exact same thing, these are really synonymous.
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It is this capital C with N and R are superscript and subscripts.
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Those really mean the same thing.
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N choose R as a binomial coefficient or C of NR.
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Sometimes people even write it as C of NR, that is not common.
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Depending on what textbook you are using or depending what your teachers preferences might be.
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You might use that notation as well.
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Let me emphasize that this binomial coefficient notation is not the same as fraction notation.
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There is no horizontal bar.
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This is as not N ÷ R, it is definitely not.
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This is a separate notation even though sometimes people mix it up with fractional notation.
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The way you calculate a binomial coefficient is using factorials.
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This is an actual fraction here.
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It is N! ÷ R! × N - R!.
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Like I have showed you in the previous lecture, where that formula comes from.
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You can figure it out yourself but that is how we calculate combinations or binomial coefficients.
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An example of that is unordered selection without replacement is when
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we are selecting 5 players for basketball team from the pool of 20 candidates,
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for an informal pickup game in the park.
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Stress here is that we are going to take 5 people, we cannot repeat the same person.
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We cannot pick the same person more than once.
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We are selecting 5 people, really 5 different people and it is like the same person more than once.
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That is why it is without replacement, you cannot pick the same person more than once.
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And it is unordered because this is an informal game in the park meaning that
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we are just going to throw 5 people out on the basketball court.
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It does not matter what order they run out there, they are just going to run out in one big group.
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This is an unordered selection.
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The number of different ways we can do that is 20 choose 5.
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Expanding that out, it is 20! ÷ 5! × 15!.
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15, I got that by looking at this N - R that is 20 - 5!.
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That would be a very large number which is why I'm not going to calculate it out.
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I’m just leaving it in factorial form.
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If you calculated that out, that is the total number of ways you can pick
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your basketball team for an informal pickup game in the park.
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Let us move on.
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Those are combinations, let us learn about permutations which are very similar
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except we are going to change unordered to ordered.
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Let us see how that is different.
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Permutations looks very similar to combinations but the difference here is that permutations,
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you are counting ordered objects.
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You are making it ordered selection.
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You are still selecting from N possibilities.
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You are still selecting R objects and you are still using without replacement.
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You are still cannot pick the same thing twice.
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But the difference here is that we are talking about ordered objects instead of unordered.
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The notation for that is P of NR.
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Sometimes people use P of NR like that but it is not common.
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The distinction here is that before we had an R! here, and here we do not have and R! anymore.
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There is no more R! in permutations.
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There was an R! for combinations.
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Otherwise, it is the same formula N! ÷ N - R!.
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Let us see a quick example of this.
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I try to make a similar to the previous example so that you can see what the only difference is.
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We are going to select 5 players for a basketball team.
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We have 20 candidates, that is all the same as before.
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The difference here is that we have fixed positions for this game.
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We are going to have a center, we are going to have a power forward, and so on.
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We are going to fill in the other positions, point guard, shooting guard, and so on.
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The difference is that now it matters who plays center and who plays power forward.
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That makes it an ordered selection because if we picked Tom to be the center and then Harry to be power forward,
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that is different from picking Harry to be center and Tom to be power forward.
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We are going to use the P formula, permutations, instead of the C formula combinations to solve this.
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The permutations is 20! ÷ 15!.
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That one actually would simplify fairly nicely.
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It would be 20 × 19 × 18 × 17 × 16 and then I do not have to keep going through 15
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because all 15 down through 1 got divided away by the 15!.
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This would be equal to 20 × 19 × 18 × 17 × 16.
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Another way to see that is, first you pick your center and there is 20 people you can pick for the center.
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And then you pick your power forward and there is 19 people left
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to choose as your power forward, and so on down to your last position.
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Maybe that is the point guard.
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At that point, there is only 16 people left, 16 choices for the point guard.
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That is how you can see that the answer is 20 × 19 × 18 × 17 × 16.
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Remember, the distinction here is that we are ordering the position.
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It matters who plays center and who plays point guard there.
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Let us see some general formulas for this.
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The key formulas when you are choosing R things from N possibilities,
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the questions you have to ask are, is it with replacement or without replacement?
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Is it ordered or unordered?
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Once you get clear on which of those categories you are in, there is a simple formula for the answer.
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Ordered with replacement, it is N ⁺R.
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Ordered without replacement, it is P of NR.
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I will just remind you that that was N!/ N - R!.
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Unordered with replacement, that is the most complicated one.
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It is N + R – 1 choose R.
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Unordered without replacement is N choose R.
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Let us go through and do some examples.
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I try to write several examples that all sound very similar to each other
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but it is designed to test your understanding of these key concepts.
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Are we talking about ordered or unordered?
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Are we talking about with replacement or without replacement?
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Let us go ahead and look at those.
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First example, we got 5 different candy bars and we are going to give them to 20 children.
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We would not give any one child more than one candy bar.
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And then the question is, is this with replacement or without?
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Is it unordered or ordered?
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And how many ways can we distribute the candy.
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What we are doing here is we got 5 candy bars, for each candy bar
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we are going to choose a child to give the candy bar to.
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We are making 5 choices here.
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We are choosing R = 5 children because for each candy bar, we choose a child to give it to.
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Each time we choose a child, there are 20 possible children we can give it to.
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That is N = 20 possible children.
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The key points here are, first of all, we do not want to give any child more than one candy bar.
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That means once I have given a candy bar to a child, that child has to leave the room.
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I do not get to pick him again to give another candy bar to.
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That means I'm choosing without replacement.
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Once that child gets a candy bar, that child is not available.
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It is not replaced as a choice for the next candy bar.
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Without replacement, that is the answer to our first question here.
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Second question is, is it ordered or unordered?
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If I choose Susan and then for the first candy bar and then Tom for the second,
00:20:31.800 --> 00:20:37.000
is that different from choosing Tom for the first candy bar and then Susan for the second?
00:20:37.000 --> 00:20:48.000
The clue to the distinction there is in the first sentence, we are giving 5 different candy bars out.
00:20:48.000 --> 00:20:54.400
Because we are giving 5 different candy bars out, let us say I’m giving out snickers and the 3 musketeers.
00:20:54.400 --> 00:20:58.200
I will give out a snickers and then the 3 musketeers.
00:20:58.200 --> 00:21:04.000
If I give the snickers to Susan and the 3 musketeers to Tom,
00:21:04.000 --> 00:21:12.200
that is different from giving Tom the snickers and Susan the 3 musketeers.
00:21:12.200 --> 00:21:13.600
These are different.
00:21:13.600 --> 00:21:16.900
They are different because they are different candy bars.
00:21:16.900 --> 00:21:26.300
These 2 distributions are different, that means that the order matters.
00:21:26.300 --> 00:21:31.800
If I'm going to give out the snickers first and then the 3 musketeers,
00:21:31.800 --> 00:21:34.600
it matters which order I pick the children in.
00:21:34.600 --> 00:21:39.800
If I pick Susan first and then Tom, they are going to get a different distribution than
00:21:39.800 --> 00:21:44.500
if I pick Tom first and then Susan.
00:21:44.500 --> 00:21:49.100
The fact that they are different candy bars means that order matters.
00:21:49.100 --> 00:21:58.000
This is an ordered selection.
00:21:58.000 --> 00:22:00.200
This is ordered without replacement.
00:22:00.200 --> 00:22:05.100
Now, I just look up on my formula table and I see ordered without replacement.
00:22:05.100 --> 00:22:10.100
If you check a couple slides back, you will see the ordered without replacement means
00:22:10.100 --> 00:22:23.600
we use P of NR which in this case is P of 25 because N is 20, R is 5.
00:22:23.600 --> 00:22:31.000
This is 20!/ 20 - 5!.
00:22:31.000 --> 00:22:36.800
Using the formula for permutations, N!/ N - r !.
00:22:36.800 --> 00:22:43.600
This is 20!/ 15!.
00:22:43.600 --> 00:22:49.700
And if you want to simplify that, you can cancel out all the factors of 1 through 15 from the 20!.
00:22:49.700 --> 00:23:01.300
You will get 20 × 19 × 18 × 17 × 16 ways to distribute this candy.
00:23:01.300 --> 00:23:05.900
The other way to think about that is, I got 5 different candy bars.
00:23:05.900 --> 00:23:11.700
I look at the first candy bar and I say there is 20 children.
00:23:11.700 --> 00:23:16.700
I can give that first bar to then I would give that first bar away and
00:23:16.700 --> 00:23:22.700
I send that child out of the room because I do not want that child to get another candy bar.
00:23:22.700 --> 00:23:26.900
Give the second bar out, I got of 19 choices for the second bar.
00:23:26.900 --> 00:23:28.500
I will send that child out of the room.
00:23:28.500 --> 00:23:34.300
I got 18 choices left and I go through and distribute all my candy.
00:23:34.300 --> 00:23:40.600
By the time I give out the last bar, there is 16 children left trying to get that last candy bar.
00:23:40.600 --> 00:23:48.100
Just to focus here on what is important, what we are trying to decide is with replacement or without.
00:23:48.100 --> 00:23:56.900
The way we know it is without replacement is that we do not want to give the child more than one candy bar.
00:23:56.900 --> 00:24:02.900
Once I choose a child, that child does not go back in the pool and he is not replaced in the pool.
00:24:02.900 --> 00:24:05.400
It is without replacement.
00:24:05.400 --> 00:24:08.500
It is an ordered selection because the candy bars are all different.
00:24:08.500 --> 00:24:13.700
It matters who gets which candy bar.
00:24:13.700 --> 00:24:21.100
I look at my chart, I see that without replacement and ordered gives me permutations of N and R.
00:24:21.100 --> 00:24:28.800
I fill in the formula for permutations, simplify it down, and that is the number of ways that I can do it.
00:24:28.800 --> 00:24:32.400
The next example is going to look very similar to this one.
00:24:32.400 --> 00:24:38.000
You want to read it very carefully and see if you notice the difference between the next example and this one.
00:24:38.000 --> 00:24:41.000
It is going to look almost the same, there is one small difference and
00:24:41.000 --> 00:24:44.300
that is going to dramatically change the answer that we get.
00:24:44.300 --> 00:24:46.500
That is really how probability questions go.
00:24:46.500 --> 00:24:50.600
Very small differences in the wording made a big difference in the answer.
00:24:50.600 --> 00:24:54.500
Let us check out the next example.
00:24:54.500 --> 00:24:58.000
We have 5 identical candy bars to distribute to 20 children.
00:24:58.000 --> 00:25:01.100
We do not want to give any child more than one candy bar.
00:25:01.100 --> 00:25:04.600
Is the selection with replacement or without? Is it ordered or unordered?
00:25:04.600 --> 00:25:08.500
How many ways can we distribute the candy?
00:25:08.500 --> 00:25:12.400
This looks almost the same as the previous example.
00:25:12.400 --> 00:25:21.200
Again, we are choosing R = 5 children.
00:25:21.200 --> 00:25:30.500
We are making 5 choices from each time we choose a child, there are 20 children hoping to get that candy bar.
00:25:30.500 --> 00:25:41.500
We have N = 20 possibilities.
00:25:41.500 --> 00:25:44.900
Is this selection with replacement or without?
00:25:44.900 --> 00:25:49.000
Remember, we said we do not want to give any child more than one candy bar.
00:25:49.000 --> 00:25:54.800
That means, once we select the child, give that child a candy bar, he has to leave the room.
00:25:54.800 --> 00:25:59.200
He does not go back in the pool of recipients.
00:25:59.200 --> 00:26:03.900
He is not replaced in the pool.
00:26:03.900 --> 00:26:12.300
Because we do not want to give any child more than one candy bar, this is without replacement.
00:26:12.300 --> 00:26:19.000
No child can have more than one candy bar.
00:26:19.000 --> 00:26:21.700
That is what that tells us right there.
00:26:21.700 --> 00:26:28.100
Is this selection unordered or ordered?
00:26:28.100 --> 00:26:32.800
Remember, last time we have 5 different candy bars.
00:26:32.800 --> 00:26:37.400
This time, we have 5 identical candy bars.
00:26:37.400 --> 00:26:40.500
That means we are giving out 5 snickers.
00:26:40.500 --> 00:26:46.300
I will give you a snickers, I will give you a snickers.
00:26:46.300 --> 00:26:52.300
What that means is, if I choose Susan to get the first snickers
00:26:52.300 --> 00:26:58.600
and then I choose Tom to get the second snickers, they both walk away with a snickers.
00:26:58.600 --> 00:27:06.100
If I choose Tom to get the first snickers and then I choose Susan to get the second snickers,
00:27:06.100 --> 00:27:07.300
they both walk away with the snickers.
00:27:07.300 --> 00:27:11.200
It is the same to them, it is the same distribution of candy.
00:27:11.200 --> 00:27:17.100
As if I had chosen Susan first and the Tom second or if I had chosen Tom first
00:27:17.100 --> 00:27:20.700
and Susan second because the candy bars look the same.
00:27:20.700 --> 00:27:27.900
If they were different candy bars, the order would matter but because the candy bars are identical,
00:27:27.900 --> 00:27:36.400
this is an unordered selection.
00:27:36.400 --> 00:27:40.000
That really depends on the wording of the problem.
00:27:40.000 --> 00:27:42.500
This problem is almost identical to the previous one.
00:27:42.500 --> 00:27:47.700
The only difference is that the candy bars are identical, instead of being different.
00:27:47.700 --> 00:27:52.400
But because of that subtle difference, we use a different formula.
00:27:52.400 --> 00:27:55.700
The formulas, I lay them all on a chart in one of the earlier slides.
00:27:55.700 --> 00:27:57.900
You can go back and look it up.
00:27:57.900 --> 00:28:07.800
The answer for unordered with replacement is, we use combinations to calculate this.
00:28:07.800 --> 00:28:15.800
The binomial coefficient C of 25 or you can also think of this as 20 choose 5.
00:28:15.800 --> 00:28:20.500
That is different notation for the same thing.
00:28:20.500 --> 00:28:28.400
This is from N = 20 and R = 5, the number of choices we are making.
00:28:28.400 --> 00:28:40.700
Remember, the way you expand a binomial coefficient or the combination formula is N!/ 5!.
00:28:40.700 --> 00:28:43.700
Let me write that as R.
00:28:43.700 --> 00:28:46.700
× N - R!.
00:28:46.700 --> 00:28:49.200
I will plug in what the numbers are.
00:28:49.200 --> 00:28:59.100
I get 20!/ R is 5! And 20 -5 is 15!.
00:28:59.100 --> 00:29:05.700
And I think what I will do is I will cancel out some of the terms from 20! With the 15!.
00:29:05.700 --> 00:29:13.200
That will leave me with 20 × 19 × 18 × 17 × 16.
00:29:13.200 --> 00:29:17.800
I’m canceling out all the numbers after that with the 15!.
00:29:17.800 --> 00:29:22.000
I will divide that by 5! because that is still in there.
00:29:22.000 --> 00:29:26.100
5 × 4 × 3 × 2 × 1.
00:29:26.100 --> 00:29:31.600
It looks like there is a lot more cancellation I could do there.
00:29:31.600 --> 00:29:35.800
I think I will go ahead and cancel the 20 with the 5 × 4.
00:29:35.800 --> 00:29:41.500
I will cancel the 18 with 3 and turn it into a 6.
00:29:41.500 --> 00:29:46.800
And then I can turn that 6 cancel that with the 2 and give me 3.
00:29:46.800 --> 00:29:52.500
I get 19 × 3 × 17 × 16.
00:29:52.500 --> 00:29:53.500
It is still a pretty big number.
00:29:53.500 --> 00:29:56.200
I think I do not want to multiply that out.
00:29:56.200 --> 00:30:05.600
But that is the number of ways that I can distribute my identical candy bars to these 20 deserving children.
00:30:05.600 --> 00:30:11.400
Let me emphasize here that this one is almost identical to the previous problem except
00:30:11.400 --> 00:30:13.300
that the candy bars are identical now.
00:30:13.300 --> 00:30:16.900
We are still making 5 choices from 20 possibilities.
00:30:16.900 --> 00:30:19.900
We are still choosing without replacement.
00:30:19.900 --> 00:30:24.900
That is because we do not want to give a child more than one candy bar.
00:30:24.900 --> 00:30:30.400
It means that after we give a child a candy bar, that child is not replaced in the pool.
00:30:30.400 --> 00:30:36.200
That child has to leave the room, cannot get another candy bar.
00:30:36.200 --> 00:30:43.000
The difference is that this is an unordered selection because the candy bars all look the same.
00:30:43.000 --> 00:30:50.500
The children do not really care who gets picked first because they are all going to get the same candy bar in the end.
00:30:50.500 --> 00:30:53.800
The candy bars all look the same.
00:30:53.800 --> 00:30:58.700
Because it is unordered, because it is without replacement, we use our combination formula.
00:30:58.700 --> 00:31:05.300
This is coming from the chart that we had on the slide a few minutes ago.
00:31:05.300 --> 00:31:07.000
I use the combination formula.
00:31:07.000 --> 00:31:12.000
These are two different notations for the same formula but they both expand
00:31:12.000 --> 00:31:19.900
into the same fraction of N!/ R! × N - R!.
00:31:19.900 --> 00:31:25.400
Plugging in N and R and then expanding and cancelling what I can,
00:31:25.400 --> 00:31:29.300
gives me the total number of ways I can distribute the candy.
00:31:29.300 --> 00:31:33.200
If you check back and compare this with example 1, it is almost the same
00:31:33.200 --> 00:31:38.100
with just one word that I changed but the answer is quite different at the end.
00:31:38.100 --> 00:31:44.200
There are fewer ways to distribute this candy than there was in example 1.
00:31:44.200 --> 00:31:46.400
We are going to keep going with this in the other examples.
00:31:46.400 --> 00:31:50.200
I’m going to make small changes but it is going to keep changing the answers.
00:31:50.200 --> 00:31:57.100
The idea is to practice the distinction between with or without replacement and ordered vs. unordered.
00:31:57.100 --> 00:32:01.200
Let us give out some more candy in example 3.
00:32:01.200 --> 00:32:06.200
We 5 identical candy bars to distribute to 20 children.
00:32:06.200 --> 00:32:08.900
We are willing to give some children more than one candy bar.
00:32:08.900 --> 00:32:15.000
If I happen to call the same child twice, that child gets 2 candy bars and that is okay.
00:32:15.000 --> 00:32:18.900
We are being asked is the selection with replacement or without?
00:32:18.900 --> 00:32:20.500
Is it ordered or unordered?
00:32:20.500 --> 00:32:25.500
How many ways can we distribute the candy?
00:32:25.500 --> 00:32:28.800
Just as before, just as with the first two examples.
00:32:28.800 --> 00:32:30.000
This is very similar to those.
00:32:30.000 --> 00:32:33.600
We have R = 5 children.
00:32:33.600 --> 00:32:39.100
We are making 5 choices because I have my 5 candy bars.
00:32:39.100 --> 00:32:47.900
Each time I’m going to pick a child and give that child a candy bar from N = 20 possibilities.
00:32:47.900 --> 00:32:59.200
Because each time I pick a child, there is 20 children to choose from.
00:32:59.200 --> 00:33:03.300
Is this selection with replacement or without?
00:33:03.300 --> 00:33:10.200
That means if I choose a child, can I choose again and give that same child a second candy bar?
00:33:10.200 --> 00:33:21.500
According to the stand with the problem, yes, they are willing to give some children more than one candy bar.
00:33:21.500 --> 00:33:25.900
Based on that, after I choose a child, that child gets to stay in the room
00:33:25.900 --> 00:33:30.600
and is still eligible to get a second candy bar.
00:33:30.600 --> 00:33:34.200
We are choosing here with replacement.
00:33:34.200 --> 00:33:44.900
That child gets replaced into the pool because that child might get to stick around and get a second candy bar.
00:33:44.900 --> 00:33:49.000
Now, our second question is, is unordered or ordered?
00:33:49.000 --> 00:33:56.200
That means if I pick Susan and then I pick Tom, is that different from picking Tom and then picking Susan?
00:33:56.200 --> 00:33:59.800
The key is to look at the candy bars.
00:33:59.800 --> 00:34:05.200
In this case, all the candy bars are identical.
00:34:05.200 --> 00:34:08.200
If I give a candy bar to Susan and then a candy bar to Tom,
00:34:08.200 --> 00:34:14.300
it is going to be the same as if I give a candy bar to Tom and a candy bar to Susan.
00:34:14.300 --> 00:34:16.000
That means the order does not matter.
00:34:16.000 --> 00:34:19.000
Either way they both get a candy bar.
00:34:19.000 --> 00:34:30.600
It is unordered, this is an unordered selection with replacement.
00:34:30.600 --> 00:34:34.200
We can figure out how many different ways there are to do this,
00:34:34.200 --> 00:34:38.900
by looking at our chart that I had a few slides back.
00:34:38.900 --> 00:34:50.700
Unordered with replacement, the formula for that is the binomial coefficient N + R -1 choose R.
00:34:50.700 --> 00:35:02.800
The notation for the same thing is C of N + R – 1 R.
00:35:02.800 --> 00:35:07.000
In this case, N + R -1.
00:35:07.000 --> 00:35:09.200
N is 20, R is 5.
00:35:09.200 --> 00:35:18.800
20 + 5 - 1 that is 24 and R is still 5.
00:35:18.800 --> 00:35:36.700
If you remember our formulas for the binomial coefficients, for combinations, it is 24! ÷ 5! and 24 - (5!).
00:35:36.700 --> 00:35:45.500
That is 24!/ 5! × 19!.
00:35:45.500 --> 00:35:53.700
And if you want to expand this out then on you can solve the 24! or at least a lot of the terms,
00:35:53.700 --> 00:35:56.800
a lot of the factors with the 19!.
00:35:56.800 --> 00:36:04.200
You will just be left with 24 × 23 × 22 × 21 × 20.
00:36:04.200 --> 00:36:07.200
And then from there on, it is 19!.
00:36:07.200 --> 00:36:11.400
It would cancel out with the 19! in the denominator.
00:36:11.400 --> 00:36:18.500
In the bottom still is the 5! 5 × 4 × 3 × 2 × 1.
00:36:18.500 --> 00:36:21.300
I guess we can keep going with that.
00:36:21.300 --> 00:36:28.700
We can cancel the 5 and the 4 with the 20 and then the 3 and the 2, that is 6.
00:36:28.700 --> 00:36:32.000
We can cancel out with 24 and get 6.
00:36:32.000 --> 00:36:44.700
We get 6 × 23 × 22 × 21 ways that we can distribute this candy.
00:36:44.700 --> 00:36:46.200
And then you can multiply these numbers together.
00:36:46.200 --> 00:36:49.800
I do not think it would be that illuminating to multiply the numbers together,
00:36:49.800 --> 00:36:53.400
that is why I’m leaving that one in factored form.
00:36:53.400 --> 00:36:59.200
Let me go back and just make sure it is clear how we got each step of that.
00:36:59.200 --> 00:37:03.000
There are two key phrases that tell you how to calculate this and
00:37:03.000 --> 00:37:04.400
they come from the wording of the problem.
00:37:04.400 --> 00:37:09.300
Probability is so subtle, you really get to read these problems very carefully.
00:37:09.300 --> 00:37:13.200
If there is anything unclear, it is often going to ask your teacher just
00:37:13.200 --> 00:37:15.400
to make sure you know what you are being asked.
00:37:15.400 --> 00:37:22.000
Because the subtleties in the wording really affect the answer.
00:37:22.000 --> 00:37:27.000
Here, we are making 5 choices because I have got 5 candy bars.
00:37:27.000 --> 00:37:33.000
Each time I have a candy bar, I’m going to choose 1 of 20 children to hand it out to.
00:37:33.000 --> 00:37:37.400
There is 20 possibilities for each one of those choices.
00:37:37.400 --> 00:37:44.300
With the replacement, that comes from the fact that I'm willing to give some children more than one candy bar.
00:37:44.300 --> 00:37:49.800
If I give all 5 candy bars to Susan, that Susan’s lucky day.
00:37:49.800 --> 00:37:52.600
We are willing to consider that possibility.
00:37:52.600 --> 00:37:54.800
We want to count that possibility.
00:37:54.800 --> 00:37:56.600
That is why we say it is with replacement.
00:37:56.600 --> 00:38:00.200
After I gave Susan the first candy bar, she gets to go back into the pool.
00:38:00.200 --> 00:38:05.400
She gets replaced in the pool and she gets to hope that maybe she can get a second candy bar too.
00:38:05.400 --> 00:38:09.800
It is unordered because these candy bars are identical.
00:38:09.800 --> 00:38:12.000
The order does not make a difference.
00:38:12.000 --> 00:38:15.300
If I gave Susan a candy bar and then Tom a candy bar,
00:38:15.300 --> 00:38:20.000
that is the same as if I give Tom a candy bar and Susan a candy bar.
00:38:20.000 --> 00:38:22.500
They candy bars look the same.
00:38:22.500 --> 00:38:26.600
They both walk out of the room with the same kind of candy bar.
00:38:26.600 --> 00:38:32.400
Because it is unordered and with replacement, if you look back in that chart from a few slides ago,
00:38:32.400 --> 00:38:37.100
we use this combination formula N + R -1 choose R.
00:38:37.100 --> 00:38:40.100
This is just a different notation for the same thing.
00:38:40.100 --> 00:38:49.500
N is 20 and R is 5, we get this and then this is my formula for combinations.
00:38:49.500 --> 00:38:57.200
Expand that out into factorials and then the 24! has a lot of factors in common with the 19!.
00:38:57.200 --> 00:39:01.600
That is why I cancel those off at this step and then I did some more cancellations
00:39:01.600 --> 00:39:06.800
to simplify the numbers a bit and got my final answer there.
00:39:06.800 --> 00:39:10.400
The 4th example is again going to look very similar to this one.
00:39:10.400 --> 00:39:14.200
I just made a very small change in the wording and you will see that
00:39:14.200 --> 00:39:19.200
it does change the answer in the original wording looks very similar.
00:39:19.200 --> 00:39:22.500
Let us go ahead and see how that one plays out.
00:39:22.500 --> 00:39:28.500
Example 4, we have 5 different candy bars to distribute to 20 children.
00:39:28.500 --> 00:39:31.800
We are willing to give some children more than one candy bar.
00:39:31.800 --> 00:39:33.200
Is this with replacement or without?
00:39:33.200 --> 00:39:34.900
Is it unordered or ordered?
00:39:34.900 --> 00:39:39.000
How many ways can we distribute the candy?
00:39:39.000 --> 00:39:51.400
Again, we are making R = 5 choices because for each candy bar, I make a choice of which child to give it to.
00:39:51.400 --> 00:39:57.200
There is 20 children and my N is 20 possibilities.
00:39:57.200 --> 00:40:02.700
20 children that each time I have a candy bar, I look at 20 possible hungry faces
00:40:02.700 --> 00:40:10.900
and decide which one I want to give the candy bar to.
00:40:10.900 --> 00:40:17.300
I have to say is this with replacement or without?
00:40:17.300 --> 00:40:22.000
The key phrase here is we are willing to give some children more than one candy bar.
00:40:22.000 --> 00:40:27.000
If I give the first candy bar to Susan, I’m willing to give another one for the second candy bar.
00:40:27.000 --> 00:40:29.100
She gets replaced in the pool.
00:40:29.100 --> 00:40:35.200
She gets to go back into the pool and hopefully get another candy bar.
00:40:35.200 --> 00:40:53.600
That means that we are working again with replacement.
00:40:53.600 --> 00:41:01.900
The next question is, is this an unordered or ordered selection?
00:41:01.900 --> 00:41:07.100
I will put just a single star there because that was the first question.
00:41:07.100 --> 00:41:12.400
The second question is, are we working unordered or ordered?
00:41:12.400 --> 00:41:18.100
The key to answering that is the fact that we have different candy bars.
00:41:18.100 --> 00:41:25.600
This time, we maybe have snickers and the 3 musketeers and maybe several other candy bars,
00:41:25.600 --> 00:41:28.600
but they are all different.
00:41:28.600 --> 00:41:34.900
If we pick Susan first and she gets the snickers, and then Tom gets the 3 musketeers,
00:41:34.900 --> 00:41:41.000
that is different from picking Tom first to get the snickers and Susan to get the 3 musketeers.
00:41:41.000 --> 00:41:44.500
That would make a difference, maybe Susan particularly likes snickers.
00:41:44.500 --> 00:41:46.700
In the first arrangement, she would be very happy.
00:41:46.700 --> 00:41:51.600
In the second arrangement, she would be upset because she did not get the candy bar she liked.
00:41:51.600 --> 00:41:57.900
The order really matters, those are two different arrangements there.
00:41:57.900 --> 00:42:02.100
We would be picking 3 more children there or possibly the same children again.
00:42:02.100 --> 00:42:04.000
The order really matters there.
00:42:04.000 --> 00:42:12.000
This is an ordered selection.
00:42:12.000 --> 00:42:17.500
This is with replacement and it is ordered.
00:42:17.500 --> 00:42:20.900
If you go back and look at the chart, how many ways are there to make 5 choices
00:42:20.900 --> 00:42:25.100
from 20 possibilities and it is ordered and with replacement?
00:42:25.100 --> 00:42:31.900
The answer from the chart, this chart was on 3 or 4 slides ago.
00:42:31.900 --> 00:42:35.000
Just scroll back through the video and you will find that chart.
00:42:35.000 --> 00:42:39.900
The chart tells us that the answer is N ⁺R.
00:42:39.900 --> 00:42:45.700
In this case, our N is 20 and our R is 5.
00:42:45.700 --> 00:42:53.100
There are 20⁵ ways of distributing the candy, that is our answer.
00:42:53.100 --> 00:42:58.700
The keywords that you want to look for in the wording are the fact that we are willing
00:42:58.700 --> 00:43:00.900
to give some children more than one candy bar.
00:43:00.900 --> 00:43:04.400
That is how we know it is with replacement.
00:43:04.400 --> 00:43:06.100
The fact that they are different candy bars.
00:43:06.100 --> 00:43:10.800
What really matters, which kid comes first and get which candy bar.
00:43:10.800 --> 00:43:12.500
That is why it is an ordered selection.
00:43:12.500 --> 00:43:18.200
We count Susan and then Tom different from Tom and then Susan.
00:43:18.200 --> 00:43:22.100
Ordered with replacement from our chart is N ⁺R ways.
00:43:22.100 --> 00:43:23.800
That is where we get the 20⁵.
00:43:23.800 --> 00:43:27.900
This one by the way is one word pretty easy to confirm the answer.
00:43:27.900 --> 00:43:32.600
If you think about it, when you got that first candy bar, that snickers bar,
00:43:32.600 --> 00:43:35.000
you look around and you see 20 hungry faces.
00:43:35.000 --> 00:43:40.300
You did make a decision about which child you are going to give it to.
00:43:40.300 --> 00:43:46.000
You have 20 possible choices for that first candy bar then you see the second hungry child
00:43:46.000 --> 00:43:50.400
or the second candy bar, and you look around at that sea of faces again,
00:43:50.400 --> 00:43:58.100
there are still 20 kids clambering for that second candy bar because the first child got to go back into the group.
00:43:58.100 --> 00:44:07.700
There are 20 possibilities to give away that second candy bar and then 20 possibilities for the third bar as well.
00:44:07.700 --> 00:44:12.100
First bar, second bar, and so on.
00:44:12.100 --> 00:44:20.400
There are 20 possibilities for each candy bar and every child gets counted for every candy bar
00:44:20.400 --> 00:44:27.000
because even if you pick a child for the first candy bar, that child can still line up again and ask for a second candy bar.
00:44:27.000 --> 00:44:32.500
You are multiplying together 25 times, we get 20⁵.
00:44:32.500 --> 00:44:37.400
That one was fairly easy to see the answer intuitively.
00:44:37.400 --> 00:44:43.800
It is also possible just to read the answer off our chart and that is what we did.
00:44:43.800 --> 00:44:45.200
We got one more example here.
00:44:45.200 --> 00:44:46.800
It is going to be different from this one.
00:44:46.800 --> 00:44:49.800
We are not going to give any more candy to any more children.
00:44:49.800 --> 00:44:53.100
I try to make it a little different and a little less obvious.
00:44:53.100 --> 00:44:56.200
But again, it is going to be the same key decisions.
00:44:56.200 --> 00:44:59.200
Is it an ordered selection or unordered?
00:44:59.200 --> 00:45:01.400
Is it with replacement or without replacement?
00:45:01.400 --> 00:45:04.400
Let us check that one out.
00:45:04.400 --> 00:45:06.100
This one is still food related.
00:45:06.100 --> 00:45:11.000
I must have been hungry when I was writing this lecture.
00:45:11.000 --> 00:45:15.300
What we are going to do is we are going to go to a restaurant and we got a big party going on at home.
00:45:15.300 --> 00:45:21.100
We are going to buy 10 pizzas from a restaurant, bring them all back home to our hungry guests in our party.
00:45:21.100 --> 00:45:26.900
It is a fairly simple restaurant, they only carry 3 kinds of pizzas.
00:45:26.900 --> 00:45:33.500
They carry cheese pizzas and pepperoni pizzas and the vegan pizzas.
00:45:33.500 --> 00:45:39.600
We want to buy 10 pizzas total and we could buy all 10 cheese or
00:45:39.600 --> 00:45:44.700
we could buy 5 cheese and 2 pepperonis and 3 vegan pizzas.
00:45:44.700 --> 00:45:48.300
We want to figure out how many different orders can we make.
00:45:48.300 --> 00:45:53.000
And in particular, is this selection with replacement or without replacement?
00:45:53.000 --> 00:45:58.200
Is it ordered or unordered?
00:45:58.200 --> 00:46:01.200
Does it matter the order in which we picked the pizzas?
00:46:01.200 --> 00:46:07.600
Finally, how many possible different ways are there to make our order from the restaurant?
00:46:07.600 --> 00:46:09.600
Let us think about that.
00:46:09.600 --> 00:46:17.700
First of all, we are choosing 10 pizzas here.
00:46:17.700 --> 00:46:19.000
We get to make 10 choices.
00:46:19.000 --> 00:46:20.400
I want the first choice.
00:46:20.400 --> 00:46:23.700
I want the first pizza to be pepperoni.
00:46:23.700 --> 00:46:25.300
I want the second one to be vegan.
00:46:25.300 --> 00:46:27.800
I want the third one to be another vegan pizza.
00:46:27.800 --> 00:46:37.700
Then I want to choose one so we get to make 10 choices here.
00:46:37.700 --> 00:46:39.600
We choose 10 pizzas.
00:46:39.600 --> 00:46:42.700
That R is the number of choices that we get to make.
00:46:42.700 --> 00:46:48.500
R is 10 pizzas.
00:46:48.500 --> 00:46:57.500
Each one of those choices, we look around and we see there is only 3 possible pizzas.
00:46:57.500 --> 00:47:06.700
Each one can be cheese, pepperoni, or vegan, from 3 possibilities.
00:47:06.700 --> 00:47:09.400
That is the N there, that is the number of possibilities for each choice.
00:47:09.400 --> 00:47:16.800
N = 3 possibilities.
00:47:16.800 --> 00:47:22.600
That kind of sets up our problem here but the two key questions we have to ask are, with replacement or without?
00:47:22.600 --> 00:47:24.900
Unordered or ordered?
00:47:24.900 --> 00:47:27.400
Let us think about replacement.
00:47:27.400 --> 00:47:32.100
After I choose, I say I want the first pizza to be cheese.
00:47:32.100 --> 00:47:36.500
Does that mean I can still pick another cheese pizza or is cheese off the table now?
00:47:36.500 --> 00:47:40.900
The answer is that I can still pick another cheese pizza.
00:47:40.900 --> 00:47:43.300
Because the restaurant can make as many cheeses I like.
00:47:43.300 --> 00:47:47.700
It can make as many pepperoni, as many vegan as I like.
00:47:47.700 --> 00:47:52.400
It is possible to have more than one cheese pizza in my order.
00:47:52.400 --> 00:47:59.800
That means it must be with replacement.
00:47:59.800 --> 00:48:00.900
How can I write this down?
00:48:00.900 --> 00:48:15.200
We can say, we can get more than one, for example cheese pizza.
00:48:15.200 --> 00:48:17.100
You are making a choice with replacement.
00:48:17.100 --> 00:48:24.600
It is not like after we choose a cheese pizza, they run out of cheese and we cannot buy another one.
00:48:24.600 --> 00:48:32.500
Cheese is still an option, even after we say I want the first pizza to be cheese.
00:48:32.500 --> 00:48:35.000
That is okay, the second pizza can still be cheese,
00:48:35.000 --> 00:48:38.600
if we happen to be great fan of cheese.
00:48:38.600 --> 00:48:40.500
Unordered or ordered?
00:48:40.500 --> 00:48:46.900
Suppose we choose 5 cheese pizzas and then 5 vegan pizzas,
00:48:46.900 --> 00:48:51.000
and we take them all home to our guests at the party.
00:48:51.000 --> 00:48:54.600
First, we can choose 5 vegan pizzas and then 5 cheese pizzas,
00:48:54.600 --> 00:48:57.300
then we take them all home to our guests at the party.
00:48:57.300 --> 00:48:59.300
Will that make any difference to our guests?
00:48:59.300 --> 00:49:02.800
No, they will be happy either way.
00:49:02.800 --> 00:49:06.700
Whatever order we choose these pizzas in, we are going to pile them all together,
00:49:06.700 --> 00:49:14.100
take it back to the party and our guests are going to choose whatever they like no matter what.
00:49:14.100 --> 00:49:26.000
We come home, we returned home with 10 pizzas and it does not really matter
00:49:26.000 --> 00:49:29.900
what order they are stacked up in.
00:49:29.900 --> 00:49:39.000
With 10 pizzas, it does not really matter what order they are stacked up in.
00:49:39.000 --> 00:49:52.200
Our guests still get to pick whichever pieces they like in no particular order.
00:49:52.200 --> 00:50:01.800
This is an unordered selection because it really does not matter when we are making our order at the restaurant,
00:50:01.800 --> 00:50:08.400
it really does not matter whether we pick 5 cheese first and then 5 vegan,
00:50:08.400 --> 00:50:11.700
or 5 vegan pizzas first and then 5 cheese pizzas.
00:50:11.700 --> 00:50:17.000
We are still going to come home with 5 cheese and 5 vegan either way.
00:50:17.000 --> 00:50:18.400
It does not matter to our guests.
00:50:18.400 --> 00:50:23.900
It does not change what our order is at the restaurant.
00:50:23.900 --> 00:50:29.400
What we have here is a selection that is unordered with replacement.
00:50:29.400 --> 00:50:34.100
If you go back and look at our chart which we had a couple slides ago,
00:50:34.100 --> 00:50:38.200
we are going to use combinations to solve that, the way you solve an unordered
00:50:38.200 --> 00:50:45.400
with replacement selection is you use N + R -1 choose R.
00:50:45.400 --> 00:50:52.800
The other notation for that is C of N + R -1 R.
00:50:52.800 --> 00:50:57.200
In this case, the R is 10 and the N is 3.
00:50:57.200 --> 00:51:03.500
10 + 3 - 1 that is 13 -1 is 12.
00:51:03.500 --> 00:51:09.100
The R was 10.
00:51:09.100 --> 00:51:14.300
Remember, the formula for combinations.
00:51:14.300 --> 00:51:24.200
That is 12! ÷ 10! × 12 -2!.
00:51:24.200 --> 00:51:26.000
This one we are going to get a lot of cancellation.
00:51:26.000 --> 00:51:31.800
12! ÷ 10! × 2!.
00:51:31.800 --> 00:51:39.700
If we expand out 12!, we get 12 × 11 and then × 10 × 9 × 8, and so on.
00:51:39.700 --> 00:51:45.500
That one cancels with the 12th or the 10! in the denominator.
00:51:45.500 --> 00:51:54.300
The rest of it cancels with the 10! And we just have a 2 × 1 from the 2! in the denominator.
00:51:54.300 --> 00:51:57.300
The 12 will cancel out the 2 and give a 6.
00:51:57.300 --> 00:52:05.100
66, 6 × 11 is 66 possible orders.
00:52:05.100 --> 00:52:11.700
This number of possible orders at our pizza restaurant actually is not big.
00:52:11.700 --> 00:52:22.700
There are 66 different ways we can distribute our orders between cheese pizzas, pepperoni pizzas, and vegan pizzas.
00:52:22.700 --> 00:52:26.500
Let us recap how we arrived at that conclusion.
00:52:26.500 --> 00:52:34.800
We are buying pizzas at a pizza restaurant and we are buying 10 pizzas which is where we are making 10 choices.
00:52:34.800 --> 00:52:39.800
10 different times we get to say I want a vegan pizza, I want pepperoni pizza.
00:52:39.800 --> 00:52:43.100
Each time we make one of those choices, there are 3 possibilities.
00:52:43.100 --> 00:52:46.400
There is vegan, pepperoni, and cheese.
00:52:46.400 --> 00:52:50.000
There are 3 possibilities, that is where we get R and N there.
00:52:50.000 --> 00:52:52.500
Is it with replacement or without?
00:52:52.500 --> 00:52:56.900
Once you pick a cheese pizza, you can pick another cheese pizza if you like.
00:52:56.900 --> 00:52:59.300
It is with replacement.
00:52:59.300 --> 00:53:01.000
Is it ordered or unordered?
00:53:01.000 --> 00:53:07.600
It do not really matter whether you order the vegan pizza first and the pepperoni pizza second, or vice versa.
00:53:07.600 --> 00:53:10.900
You are still going to come home with a vegan and a pepperoni pizza.
00:53:10.900 --> 00:53:13.100
It is unordered.
00:53:13.100 --> 00:53:18.600
You just look at the chart that we have a few slides back in the introductory slides.
00:53:18.600 --> 00:53:25.300
I gave you a little chart with replacement, without replacement, unordered, ordered, and then formulas for each one.
00:53:25.300 --> 00:53:33.100
The formula for with replacement and unordered was N + R -1 choose R.
00:53:33.100 --> 00:53:38.600
When you plug in R = 10 and N = 3, you will get 12 choose 10.
00:53:38.600 --> 00:53:44.200
That is 12!/ 10! ×,
00:53:44.200 --> 00:53:45.500
There is a small mistake there.
00:53:45.500 --> 00:53:50.200
This is what happened to recap because that 2 should have been 10.
00:53:50.200 --> 00:54:00.900
I did not write on the next step or 12 -10 is 2!.
00:54:00.900 --> 00:54:07.000
12 - 10 that is why I got a 2! on the next step.
00:54:07.000 --> 00:54:13.600
The calculations are still right because it is this step, it was just a little mistake right there.
00:54:13.600 --> 00:54:17.500
12!/ 10! × 2!.
00:54:17.500 --> 00:54:24.600
When you write those out, all the factors of 12! Or most of them cancel with the factors of 10!.
00:54:24.600 --> 00:54:34.800
That is why I did not write the 10! Factors here because they all got canceled with the 10! part of 12! up here.
00:54:34.800 --> 00:54:39.400
Left me with just 2 × 1 in the denominator from the 2!.
00:54:39.400 --> 00:54:45.000
12 × 11 in the numerator cancel the 12 and the 2 and get 6 × 11
00:54:45.000 --> 00:54:53.500
and you get 66 possible ways to make our pizza order at this pizza restaurant.
00:54:53.500 --> 00:54:56.800
That is the end of this lesson on making choices.
00:54:56.800 --> 00:55:02.600
Let me emphasize that this is all about reading these problems very carefully and
00:55:02.600 --> 00:55:06.500
deciding is this an ordered choice or an unordered choice?
00:55:06.500 --> 00:55:09.800
Is this a choice with replacement or without replacement.
00:55:09.800 --> 00:55:13.100
We will do the same thing when you study your own probability problems that have
00:55:13.100 --> 00:55:16.100
to do with making choices and counting things.
00:55:16.100 --> 00:55:20.000
You will ask the same questions each time, ordered or unordered?
00:55:20.000 --> 00:55:22.700
Replacement or without replacement?
00:55:22.700 --> 00:55:27.800
And then once you have answered those questions, we have this nice chart of all the different formulas.
00:55:27.800 --> 00:55:35.800
You just drop it into a chart, you get a formula and then you can calculate the number of ways to make your choices.
00:55:35.800 --> 00:55:38.000
That is the end of this lecture on making choices.
00:55:38.000 --> 00:55:42.400
This is part of a larger series of lectures on probabilities.
00:55:42.400 --> 00:55:49.800
I hope you will stick around and sign up for the other lectures here on www.educator.com on probability.
00:55:49.800 --> 00:55:56.100
We got all kinds of good stuff divided into all kinds of categories.
00:55:56.100 --> 00:55:59.200
We are going to help you get through your probability course.
00:55:59.200 --> 00:56:00.100
Thank you very much for joining me.
00:56:00.100 --> 00:56:03.000
My name is Will Murray and this is www.educator.com.