WEBVTT mathematics/statistics/son
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Hi and welcome to www.educator .com.
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Today we are going to be talking about probability distributions.
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We are just going to start talking about them.
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So far we have covered sampling method as well as a little bit about basics of fundamental of probability.
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In probability distribution we are going to be playing those two ideas together and work on that so we can get what we want out of them.
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Because of that we need to talk about what sampling and probability cannot do.
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They cannot tell us something and because of that we need probability distribution and
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that is going to be the solution to what sampling and probability alone cannot accomplish.
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With the probability distribution we are going to be talking about random variable and what that means and the expected value is.
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We know a little bit about sampling and a little bit about probability, basically in sampling we know that
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what we are trying to do is estimate some variable by taking the sample of the population.
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Here is the population of all the people we are interested in and we take a sample out and
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we look at those individuals to try and estimate what that variable is like in the population.
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We know a little bit about the probability now in a particular population, we know the chance of getting a specific sample.
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What is the probability of getting this particular sample.
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You might want to ask yourself what is the relationship between these two things.
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It is where we are getting the sample and we are looking at the mean and the standard deviation of some particular variable in order to estimate that variable.
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In probability, what we are doing is it is not that we are actually taking a sample but we have a population.
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We are trying to think about all the different kinds of samples, the sample space and then calculate the probability of the specific sample.
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In one we are actually taking the sample and the other we are not actually taking a sample.
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Here there are some limitations because even if we try to take a random sample, one that is not biased in some way.
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Even though we try to take a sample we do not necessarily know how close that sample is to the population.
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Here we have taken a sample but we know the probability of getting that particular sample.
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We are still missing one piece of the puzzle, neither sampling nor probability helps us know the actual characteristics,
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Things like shape and spread of the population.
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We can know how likely this sample is given the population.
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We could take our sample out and summarize that sample.
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But we do not know how to understand the population if we do not already know what it is.
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What do we do?
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Here is where the probability distributions are going to come in.
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In probability distributions, what we are going to do is take this population and look at the distribution of all the different possible samples and get their probabilities.
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This way it is not that we are calculating one probability for one particular sample but we are calculating all the possible sample and all the probability that go along with it.
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That is the probability distribution.
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We are no longer dealing with single probability or single samples but we are looking at the universe of all the sample
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and the probability that go with them that might happen from a low population.
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It is not necessary to know or we definitely know that this is a population but it could be a theorize population like a fair sighted point.
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Those could be theoretical model.
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That is the known population, we have sense of the population and from that we win from it,
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all the possible sample and all the possibilities that go with the sample.
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Here is why we want to do that.
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First we use probability, the fundamental principles of probability to figure out what samples from real populations look like.
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We would want to do it from one sample at a time now we are going to do it for all the possible samples.
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We figure out what the distribution of samples looks like then we take a sample from a different unknown population.
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Here we have known population and now we get all samples and corresponding probability of those samples.
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Then we have this unknown one, unknown population, and we take a sample.
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What we do is we compare this sample to the results of the samples from the known population.
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Why this sample like this is our theorize population?
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Why is this sample unlikely in this universe of all possible samples from such a know population.
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If this sample is highly like given that we know that this is probability then we could say these two guys are similar.
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But if this sample is very unlikely then maybe we could say perhaps these two population are not similar.
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This one is different from this known population.
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That is the insight, we are going to use these probability distributions in order to help us figure out samples
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that come from population to try and figure out what the unknown population is like.
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It helps us to find what a probability distribution is.
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These are all the samples that can be drawn from an a known population and their corresponding probabilities.
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When you think about probability distributions, I also want you to think of what it is called a sampling distribution.
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What it is really is, it is not a distribution of a single entity or people or companies.
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It is not a distribution of single data point.
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Instead it is a sampling distribution.
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It is a distribution made up of a whole bunch of samples.
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Now that we know what probability distributions are, for and why we need them so desperately.
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Let us try to smoothen out the road from probability to probability distributions.
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In probability we learn about sample spaces.
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Remember when we are talking about sample spaces, we are talking about all the different possible samples that we could have.
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When we talk about flipping two coins and we look at what is the probability of getting 2 heads in a row?
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We created these sample spaces of all the possible different outcomes.
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For instance, if we look at the sample space for 2 dice we might look at something like this, dice 1, 2.
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We might have 1, 2, 3, 4, 5, 6 and 1, 2, 3, 4, 5.
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If we look at the sample space that would be 1-1, 1-2, 1-3, 1-4, 1-5, 1-6.
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Or it might be 2-6, 3-6, 4-6, 5-6, 6-6.
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If you fill in all the different possible outcomes there are 36 possible outcomes.
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If you fill in the rest of this table you could see the entire sample space.
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Each of these roll of a dice presuming that this dice are fair, each of these we have a probability of 1 out of 36.
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The likelihood of this out of all the possible outcomes is 36.
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That would be looking at the sample space.
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That is the probabilities of all the different outcomes.
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When we talk about probability distributions, often we are talking about just one number and the distribution around that one variable, whatever that variable is.
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Here notice that we have 2 numbers but we may want something like the sum of 2 dice.
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Here the sum would be 2 and the sum would be 3 and the sum would be 4.
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Maybe we want the probability distribution that looks something like this.
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Here I will put all the different possible sums.
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I may not have room for this so I might draw them separately.
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For instance, can I have a sum of 0?
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No, I cannot have a sum of 1 either.
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If I roll 2 dice the lowest possible sum I could have is 2, a 1 and a 1.
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I could have a sum of 2, 3, 4, 5, 6, 7 but I could also have, I will continue this on this side.
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It will also have sums of 8, 9, 10, 11, 12.
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I cannot have any sum higher than 12 because the 6 and 6 on each dice is the highest possible sum I could have.
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My probability distribution would be what is the probability of the sum of 2?
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That would be only one outcome out of all 36 outcomes would give you that sum of 2.
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That would be 1 out of 36.
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The same thing with 12, the only possible combination that I could get a 12 is this 6 and 6 right there.
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That would be also 1 out of 36.
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Now let us think about 3, how many rolls of the dice could possibly add up to 3?
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1 and 2 but also 2 and 1.
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Any numbers higher than that is not going to work.
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Here we would write 2/36 because 2 different rolls could give us the same sum.
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Notice that not all of the probabilities are equal.
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This one is less probable and this is more probable.
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What about 4?
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For 4, now we are starting to get into some bigger numbers here.
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1 and 3 will work but so is 2 and 2, as well as 3 and 1.
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That is going to be 3/36.
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Notice the pattern here so far?
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Also notice the pattern here is going like this and like this.
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Let us continue that pattern and see if it is correct.
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Our prediction would be that the sum for 5 would be 4/36 and let us see if that is true.
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If the sums combine and so thus 4 and 1.
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Let us see if these diagonals also do.
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These diagonals would be 2 and 3, as well as 3 and 2.
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Again, we know we could see our rule generalizing here.
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That is going to be 4/36.
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Actually this ends up working all the way up to 7.
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As 7, this would be going to be the longest diagonal.
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I will that in right here, 6 and 1, but also 5 and 2, 4 and 3, 3 and 4, 2 and 5, and also 1 and 6.
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All 6 of these here on the diagonal will add up to a sum of 7.
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That is why in games like stocks 7 is the highest probability rule.
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Is another game where 7 is the highest probability roll that is why it is not allowed.
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Here, you will see that what happens after these diagonal is that it becomes smaller again but in a perfectly symmetrical way.
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What goes up? 1, 2, 3, 4, 5, 6 comes down 5, 4, 3, 2, 1.
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That is because you are just finding those diagonals but now it is going from big to smaller and smaller then you will end up here the 1/36.
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This is probability distributions for us to start with because it is easy for us to see how we got these probabilities from these sums.
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This is a relatively small sample space and a small probability distribution we could fill in.
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There are going to be bigger spaces lined up each knot necessarily calculated by looking at the different combination and
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we will look at some of the algorithms and theorems that have been developed in order that we could have shortcuts so that we do not have to look at the entire sample space.
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But I want you to know where this comes from.
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It all comes from the sample space and looking at the probabilities.
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In the previous probability distribution we looked at the sum of the probability distributions of two dice.
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In that case, the random variable of one single variable that we are interested in on this probability distribution is the sum of the two dice.
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That is one example of a random variable.
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It is the thinking here, the thing that you are finding the probability of, the one variable.
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It does not have to be sum, it could be something else.
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For instance, in other games you might be interested in the probability distribution of the greater of the 2 dice.
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Here we have another probability distribution where instead of the sum I am looking for the probabilities of those sums.
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We choose the greater number and the probability of the greater number.
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Here I have just rewritten what I have written on the previous page except nice and neatly.
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One thing I want you to notice is if you add up all of these probabilities they add up to 1.
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That is good because this shows me that we have covered the entire scan of the different outcomes as well as the different probabilities.
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There is no part of the sample space that have not has been untouched.
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You have touched all of it.
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Here we see once again as the sums go up 7 will be the highest probability roll of the highest probability combination.
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It starts going down after 7.
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2 and 1 is perfectly symmetrical so that these sides are just the same and 7 is the mirror point.
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If we are looking at we roll some dice and we are looking at what is the probability that the greatest number there out of the two is 1?
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There is only one case of that, when it is 1 and 1 the greater number is 1.
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That is 1/36.
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But notice that this does not goes up and down because here what is the probability that when you roll a dice that the greatest number there will be 2?
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That is only going to happen in case of 1 and 2, 2 and 1, 2 and 2.
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That is 3/36.
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Let us get down to 6.
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6 is the highest number so it is going to be the greatest number at the time where all of the combinations that are like 6 and 1, 6 and 2, 6 and 3, 6 and 4.
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As well as 1 and 6, 2 and 6, 3 and 6.
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6 is frequently the highest number but we do not count 6 and 6 twice which is why it is 11/36.
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Even so we add all of the probabilities we have a total of 1 and that shows me we have covered the entire space.
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These are samples of probability distributions of two different random variables.
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One is the random variable of sum and the other is the random variable of greater number.
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We could have a random variable like which is the product, two numbers multiplied together.
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We could have all sorts of different random variables that we are interested in.
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The fact is that we think upon which the probability distribution is being decided.
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That is the thing you want to find the probabilities of.
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Now we know the variables and now we know the probability distributions, what is expected value?
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Here is the thing, once you a probability distribution like this, this is that rolling two dice and what is the probability of the sums?
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Here once you have this probability distribution of the sums, that is our random variable, and we often call our random variable X.
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We could also call it I or M or Y but we call it big letters.
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It would be handy to know the mean of this probability distribution because every distribution has a mean.
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It would be nice to know what the mean of this distribution is.
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That is often called the expected value, the mean of the probability distribution or the mean of the sampling distributions.
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The reason it is called the expected value is this.
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Over time if you keep rolling two dice over and over again, this probability should emerge, the law of large numbers.
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Over 10,000,000 rolls of two dice, we should see the frequencies corresponding to this probability.
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It would be nice to know what we should expect even without rolling a dice 10,000,000 times.
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It would be nice to know what the mean would be on average.
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In order to find the expected value, what we are doing here is what we did average because these two is not quite as frequent as the 7.
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These two should not count as much as the 7.
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This 12 although it is a bigger number it should not count as much as 7 because it is only going to have a small portion of a time.
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What we are doing is we are going to not only pick the sum and put it in our estimate but we are going to weight it by how likely we need it.
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If it is very unlikely it only contributes a little bit but it is very likely then it contributes a lot to the expected value.
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Out of those contributions you should be able to see what is the expected value over time?
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What is going to be the most likely value of this random variable?
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In order to find the expected value, you will often see it notated like this, the expected value of x, this random variable.
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Another notation for this is μ sub X.
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The reason why we think of it as a μ is that its probability distributions are theoretical distributions.
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Remember, theoretical distributions are more like populations than samples.
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It is μ sub X, it is not just μ.
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If it is only μ it would be the population.
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When you see μ sub X this means this is the expected value.
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This can be easily found by taking the sum from i(whatever your x are).
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In this case this is my x because the sum is my random variable × how likely it is.
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We need to do this for however many x we have.
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Here what we see is that if x contributes a lot because the probability is very high then this is going to be a bigger number.
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If x only contributes a little bit its value is going to be a little bit diminished.
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When we add all of those up, we hear the course of who is the loudest?
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That is our expected value.
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Some people contribute a lot and some people contribute a little bit when we add all of them up what we find is the true story.
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In order to find this, it is going to be helpful to use Excel.
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If you want to download the available Excel file I have put in all the sums as well as the probability.
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I have just put in 1/36 but Excel will put it in decimal form for you.
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What we want to do is you want to think of this multiply together as the contribution of how much is it contributing to the expected value?
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In order to do that, we would take this x and multiply it by the probability of the x.
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This 2 contributes a little bit but the 7 contributes quite a bit.
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It is much bigger than this 2.
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It goes down again where the 12 only contributes a little bit.
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In order to find expected value we add all of these up.
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We add all of these contributions up.
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Let us see what is left.
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Our expected value or the μ sub X is 7.
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Over many rolls of the dice what we will see is on average the expected value is 7, that is the mean of this probability distribution.
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That expected value.
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Now let us move on to some examples.
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You have all the tools of the game.
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In example 1, at the state fair you could play a fish for cash, a game of chance that costs $1 to play.
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You blind the fish out of cart that has a dollar amount that you have won from a giant fish bowl.
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The games have these probabilities of winning posted on the wall.
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Is it worth playing this game?
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Here we the winnings that you could potentially earn.
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You could win from $1 all the way up to $900.
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Here are your chances of winning.
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You have 1 out of 10 chance of earning back your $1 and you have 1/120,000 of winning $900.
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You are getting a lot of bang for your buck.
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Is it worth playing this game?
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I have put all of this information on example 1 tab, these are the winnings and I just put them in here and Excel has changed it in decimal form.
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The first thing you want to do is check whether all of this probability actually add up to 1.
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Remember these are posted on the wall.
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Even out there they are telling you the whole story.
00:30:10.400 --> 00:30:24.000
Let us sum these probabilities up and make sure that we cover the entire space of winnings and probability of those winnings.
00:30:24.000 --> 00:30:35.500
These probabilities only cover 22% of that probability space and there is the probability of 1 a 100% total.
00:30:35.500 --> 00:30:37.200
Let us think.
00:30:37.200 --> 00:30:41.700
What must be missing?
00:30:41.700 --> 00:30:48.000
I think the game probably does not advertise that you count the probability of winning that game.
00:30:48.000 --> 00:30:52.600
There is that probability and they are probably not just telling you.
00:30:52.600 --> 00:30:58.700
Let us see.
00:30:58.700 --> 00:31:11.000
I am going to take all of that and add that over here.
00:31:11.000 --> 00:31:34.300
In this row I am going to put 0 as the potential winning and the probability of getting 0 should be 1 – the sum of all the rest of the possible outcomes.
00:31:34.300 --> 00:31:37.800
That is almost 78%.
00:31:37.800 --> 00:31:51.500
You have likely to win nothing but once we do this, if we add up all of these including the 0 now we should have a total of 1.
00:31:51.500 --> 00:31:54.400
That is good and we want that.
00:31:54.400 --> 00:32:04.700
This shows us this is now a complete probability distribution of the random variable winnings.
00:32:04.700 --> 00:32:10.700
What is the contribution of each of these winnings to this expected value?
00:32:10.700 --> 00:32:21.200
Over many plays of this game, what is the average winning?
00:32:21.200 --> 00:32:39.200
Let us look at each one contribution and then multiply that out and we sum this up.
00:32:39.200 --> 00:32:44.800
When we look at this sum here, you will see $.60 as the expected value.
00:32:44.800 --> 00:32:49.700
Even though there is this chance of winning $900 as well as $.60.
00:32:49.700 --> 00:33:01.900
It turned out that on average if you play the game over and over again, the average winning is going to be about $.60.
00:33:01.900 --> 00:33:20.900
You can win $.60 on average so the expected value of winnings or you can write it as μ (w) is $.60.
00:33:20.900 --> 00:33:25.100
Is it worth it playing this game?
00:33:25.100 --> 00:33:35.900
That costs you a buck to play the game so if you play the game over and over again, let us say a 100 times that is going to cost you $100.
00:33:35.900 --> 00:33:47.100
But over time if you multiply this by a 100 you are going to win $60 for every $100 you spend.
00:33:47.100 --> 00:33:49.800
That is not worth it.
00:33:49.800 --> 00:33:52.900
Over the long call you are going to be losing money.
00:33:52.900 --> 00:34:02.700
It does not matter that much if you are just going to play the game once.
00:34:02.700 --> 00:34:12.800
It is not going to tell you whether that part that you picked is going to be $.60 because remember there is no possible way you could earn $.60.
00:34:12.800 --> 00:34:20.400
This is not what that means, probably it is going to be 0 because it is 79%.
00:34:20.400 --> 00:34:24.200
4 out of 5 times you are going to be drawing 0.
00:34:24.200 --> 00:34:28.900
It is not about any 1 particular turn.
00:34:28.900 --> 00:34:33.100
You might think that is useless.
00:34:33.100 --> 00:34:34.900
Actually it is not.
00:34:34.900 --> 00:34:43.400
Move yourself around and put yourself in the seat of the guy who owns this game, fish for catch.
00:34:43.400 --> 00:34:45.600
You want to know the expected value.
00:34:45.600 --> 00:34:47.800
You are the owner of the game.
00:34:47.800 --> 00:34:57.400
You want to know on average, all of these different people are going to play are you going to be losing money at the end of the day or you are going to be grateful you did not?
00:34:57.400 --> 00:35:13.100
What the order of this game would say is this is a good game because if people play even though I will lose $.60 for every $1 roughly on average, I will be gaining $.40.
00:35:13.100 --> 00:35:15.900
That is my profit.
00:35:15.900 --> 00:35:22.200
This is not really for just single people.
00:35:22.200 --> 00:35:31.600
We are not trying to predict single events but we are trying to predict events over time and over many examples.
00:35:31.600 --> 00:35:33.900
Here is example 2.
00:35:33.900 --> 00:35:46.000
According to our recent government report only 16% of occupants in trucks wear seat belts, supposed you randomly sample 3 occupants of trucks
00:35:46.000 --> 00:35:55.600
what is the random variable and which doubt that this 16% estimate if not of the 3 is wearing seat belts?
00:35:55.600 --> 00:35:57.400
Let us see.
00:35:57.400 --> 00:36:04.400
In order to create a probability distribution we want to decide on what the random variable should be.
00:36:04.400 --> 00:36:09.700
We eventually want to know what if none of the 3 wear seat belts?
00:36:09.700 --> 00:36:13.200
Maybe we want to know how many seat belts and passengers?
00:36:13.200 --> 00:36:19.500
Seat belt and passengers.
00:36:19.500 --> 00:36:22.700
We have 0 seat belts and passengers that is our none of the 3.
00:36:22.700 --> 00:36:25.300
We could have 1, 2, or 3.
00:36:25.300 --> 00:36:37.800
We also want to know what is the probability of these seat belted passengers?
00:36:37.800 --> 00:36:40.700
Now we want to figure this out.
00:36:40.700 --> 00:36:49.200
If this was like head or tails, all of these combinations we have equal probability but
00:36:49.200 --> 00:36:55.900
this all have equal probability because only 16% of occupants of trucks say they wear seat belts.
00:36:55.900 --> 00:37:07.500
Although it is useful to have the sample space this is not going to be enough.
00:37:07.500 --> 00:37:10.300
First let us just look at the sample space.
00:37:10.300 --> 00:37:25.500
Here is 1, 2, and 3, and I will put in s for seat belt and n for no seat belts.
00:37:25.500 --> 00:37:41.500
Half of these are seat belted and half are not.
00:37:41.500 --> 00:37:53.600
Here is our sample space and we see that the 3 people all wear seat belts and that is 1/8 of the sample space.
00:37:53.600 --> 00:37:58.500
This is also 1/8 of the entire sample space.
00:37:58.500 --> 00:38:06.300
The story is not that simple because that would only be if wearing a seat belt and not wearing a seat belt or equally likely.
00:38:06.300 --> 00:38:08.800
That is not the case.
00:38:08.800 --> 00:38:16.500
These are independent events so we could use the multiplication rule.
00:38:16.500 --> 00:38:22.400
Let us say we have these 3 spaces what is the probability that this guy here is not wearing a seat belt?
00:38:22.400 --> 00:38:23.700
That is going to be 84%.
00:38:23.700 --> 00:38:36.400
16% wearing a seat belt and the other side of that point is 84%.
00:38:36.400 --> 00:38:44.600
That is the probability for each of these seat belts because think of them as independent events.
00:38:44.600 --> 00:39:04.400
For number 3 where all 3 people are nice seat belt wearing, well abiding citizens in this truck that is going to be 16³.
00:39:04.400 --> 00:39:06.600
These are a little more complicated.
00:39:06.600 --> 00:39:12.100
Here let us think about this, what is the probability that one person is wearing a seat belt?
00:39:12.100 --> 00:39:23.800
This got one of the seats but these people are not.
00:39:23.800 --> 00:39:30.100
This is not much more likely also in the sample space.
00:39:30.100 --> 00:39:32.100
There are 3 of these.
00:39:32.100 --> 00:39:40.200
Here is one, here is another one, there is another one.
00:39:40.200 --> 00:39:51.600
As we are using the addition rule, we would add these 3 times but we could also just multiply it by 3 to make it easier.
00:39:51.600 --> 00:39:55.700
What about if 2 people are wearing seat belts and 1 person is not?
00:39:55.700 --> 00:40:07.500
That would be .16 for the 2 people wearing seat belt that is the probability that
00:40:07.500 --> 00:40:16.500
2 people are wearing seat belts × .84 the probability that they would not be wearing seat belts.
00:40:16.500 --> 00:40:23.600
Here we see that there are also 3 cases.
00:40:23.600 --> 00:40:31.500
We could get the addition rule and add these 3 times but we could just multiply it by 3.
00:40:31.500 --> 00:40:36.100
Once we do that then we could see the actual probability.
00:40:36.100 --> 00:40:39.900
Here we do not need to multiply by anything because there is only 1.
00:40:39.900 --> 00:40:45.700
It is like multiplying by 1.
00:40:45.700 --> 00:41:06.100
I am just going to calculate this in my Excel file so I could put in .84³ so that is .84 × .84 × .84.
00:41:06.100 --> 00:41:19.100
That is 59.27%.
00:41:19.100 --> 00:41:28.100
This should be .5927.
00:41:28.100 --> 00:41:50.500
We could do the other one just to finish it up 2 × .1, 3 × .16 × .84 × .84.
00:41:50.500 --> 00:42:03.300
The next one is 3 × .16 × .16 × .84.
00:42:03.300 --> 00:42:08.300
The last one is .16³.
00:42:08.300 --> 00:42:17.800
We see that it is very likely where all 3 of them are not wearing a seat belt.
00:42:17.800 --> 00:42:31.000
It is less likely than this but still pretty likely about 34% of the time that your sample have 1% wearing seat belts.
00:42:31.000 --> 00:42:35.000
Less likely that two people would be wearing seat belt that is only like 6%.
00:42:35.000 --> 00:43:01.200
If there is less than 1% chance that all 3 people will be wearing a seat belt.
00:43:01.200 --> 00:43:07.600
Would you tend to doubt the 16% estimate if none of the 3 were wearing a seat belt?
00:43:07.600 --> 00:43:19.500
We know that if we took a sample 59% at the time that sample will have 0% wearing seat belts.
00:43:19.500 --> 00:43:28.700
It is likely that sample is consistent with this estimate of 16% of occupants in trucks do not wear seat belts.
00:43:28.700 --> 00:43:31.800
We would not necessarily doubt that 16% occupants.
00:43:31.800 --> 00:43:39.900
We might still be wrong but we do not have reason to doubt it.
00:43:39.900 --> 00:43:47.300
Here is example 3, Apple is going to get you to buy the Apple care warranty for $250 for your laptop.
00:43:47.300 --> 00:44:02.100
If you buy this Apple care thing you will get unlimited number of free repairs but if you do not you must pay $150 per repair.
00:44:02.100 --> 00:44:06.600
You find this information below on the web.
00:44:06.600 --> 00:44:13.500
What is the random variable here and according to this data is this worth getting the warranty?
00:44:13.500 --> 00:44:19.800
Here you want to predict how likely it is going to be that I am going to need a bunch of repairs?
00:44:19.800 --> 00:44:35.200
If I need only 1 repair or 0 repair the I probably should not buy this warranty thing but if I am going to need to get repair frequently then I probably want to buy the warranty.
00:44:35.200 --> 00:44:42.400
Here we could see the probability of the repairs listed here and here are the repairs.
00:44:42.400 --> 00:44:48.600
This probability distribution has the random variable of repairs.
00:44:48.600 --> 00:44:52.700
According to this data is it worth getting this warranty?
00:44:52.700 --> 00:44:59.400
One thing we want to do is to look and see what the expected value is.
00:44:59.400 --> 00:45:10.300
On average, over million people are buying these laptops what is likely the value of repairs.
00:45:10.300 --> 00:45:14.600
How many repairs are we going to need on average?
00:45:14.600 --> 00:45:25.100
I have put this data on this table here in example 3, it is the same table.
00:45:25.100 --> 00:45:31.600
It would be nice to find the expected value of x.
00:45:31.600 --> 00:45:35.300
In this case x is repairs.
00:45:35.300 --> 00:45:49.200
In order to find that, we want to find the contribution of each value of this random variable.
00:45:49.200 --> 00:45:52.800
It is this times how likely it is.
00:45:52.800 --> 00:46:09.300
0 contributes more than 4 because it is much more frequent that people need only 0 repairs.
00:46:09.300 --> 00:46:12.900
Then we add these all up.
00:46:12.900 --> 00:46:16.700
Remember our expected value is repairs.
00:46:16.700 --> 00:46:23.800
We on average will expect about 1.1 repairs.
00:46:23.800 --> 00:46:29.900
No one person will need 1.1 repairs they might need 1 repair or 2.
00:46:29.900 --> 00:46:34.100
On average, it will average up on 1.1.
00:46:34.100 --> 00:46:37.900
Remember this is on repairs, we do not know how much it is in dollar amounts.
00:46:37.900 --> 00:46:47.500
If we want it to know in dollar amount, we know that each repair if we identify the warranty will cost $150.
00:46:47.500 --> 00:46:52.400
That will be just this times $150.
00:46:52.400 --> 00:47:01.900
On average we would expect to spend $155 is that going to make Apple care worth it?
00:47:01.900 --> 00:47:12.200
I will say according to this data it is highly likely that you will need an average 1 repair.
00:47:12.200 --> 00:47:22.400
It is probably not worth getting Apple care.
00:47:22.400 --> 00:47:29.500
Here is example 4, there are 5 television shows you want to watch before your class that will begin 74 minutes later.
00:47:29.500 --> 00:47:38.900
If you randomly pick 2 shows to watch out of this 5, what is the probability that you will have finish watching both shows before your class?
00:47:38.900 --> 00:47:43.500
It is helpful here to keep in line the questions even though it is not asked here.
00:47:43.500 --> 00:47:45.700
What is the random variable?
00:47:45.700 --> 00:48:01.000
Do not be tricked because here it shows the shows A, B, C, D, and E, and it shows the minutes how long each shows is.
00:48:01.000 --> 00:48:06.100
It does not show you the probability distribution at all.
00:48:06.100 --> 00:48:10.300
In fact it is not like that there is a probability distribution for A.
00:48:10.300 --> 00:48:12.000
How like is A?
00:48:12.000 --> 00:48:16.600
Equally like as B, C, and D if we just pick randomly.
00:48:16.600 --> 00:48:25.300
What we want to know is how many minutes 2 shows will be when added together?
00:48:25.300 --> 00:48:32.500
Picking 2 shows and getting the sum of those 2 shows.
00:48:32.500 --> 00:48:45.800
Eventually what we want to have is sum of the length of 2 shows.
00:48:45.800 --> 00:48:59.200
We want all the different combinations likely AB, AC, AD, and all their sums and then the probabilities of those sums.
00:48:59.200 --> 00:49:12.200
That is going to take more room than what we have here so if you pull out this Excel file and go to example 4.
00:49:12.200 --> 00:49:15.300
I have put this information on here.
00:49:15.300 --> 00:49:20.500
What you want to do is first start off by just looking at the sample space.
00:49:20.500 --> 00:49:24.000
Let us start with the sample space.
00:49:24.000 --> 00:49:26.700
The sample space might look something like this.
00:49:26.700 --> 00:49:35.700
First we want all of the combinations with A as the first show we picked.
00:49:35.700 --> 00:49:39.900
B will be the second show.
00:49:39.900 --> 00:49:42.700
C will be the second show.
00:49:42.700 --> 00:49:44.700
Here we have D as the second show.
00:49:44.700 --> 00:49:46.600
Maybe it is helpful if we label these.
00:49:46.600 --> 00:49:59.300
First show and then first show × second show.
00:49:59.300 --> 00:50:02.100
We have all these different things.
00:50:02.100 --> 00:50:11.600
Later what we can do is get the sum by adding this and this.
00:50:11.600 --> 00:50:13.400
We could do that later.
00:50:13.400 --> 00:50:15.600
Let us create the sample space.
00:50:15.600 --> 00:50:32.200
The next combination should be all the combinations with B as the first show and A will be the second show.
00:50:32.200 --> 00:50:36.000
C, D, E as the second show.
00:50:36.000 --> 00:50:54.900
Next we have all the combinations where C as the first show and then A and B are the second show D and E are the second show.
00:50:54.900 --> 00:51:06.200
Let us go down to D being the first show and you could have A or B as the first show.
00:51:06.200 --> 00:51:14.500
C as the first show, A and B as the second show, C as the second show, or E as the second show.
00:51:14.500 --> 00:51:21.700
Presumably you do not want to watch A after you just watched A.
00:51:21.700 --> 00:51:34.600
Finally, we have E as the first show, A, B, C, D as the second show.
00:51:34.600 --> 00:51:37.500
We have all these different shows.
00:51:37.500 --> 00:51:47.400
All the combinations and we could quite easily calculate the sums.
00:51:47.400 --> 00:51:51.400
These are all the sum total minutes.
00:51:51.400 --> 00:52:15.600
What we want to do is sort this by sum because eventually we want a probability distribution that will use sum.
00:52:15.600 --> 00:52:25.300
Here I wanted to sort by sum because eventually we want to have probability distribution that have just these sum on it.
00:52:25.300 --> 00:52:31.100
We do not care if 63 was created by watching B and D or D and B.
00:52:31.100 --> 00:52:37.800
We just care that it adds up to 63 minutes.
00:52:37.800 --> 00:52:43.300
I am going to go over here and start my probability distribution.
00:52:43.300 --> 00:52:48.600
Here I will put the sum and I have a couple of different sums here.
00:52:48.600 --> 00:53:10.100
For instance I have 63 that is the sum that come up, 65 comes up, 68 comes up, 70, 71, 73, 75, 78.
00:53:10.100 --> 00:53:17.800
I want to know how likely it is 63 given all of these potential different combinations.
00:53:17.800 --> 00:53:39.800
These are all equally probable because each show has 1 out of 5 chance of being picked as the first show and 1 out of 4 chance being picked as the second show.
00:53:39.800 --> 00:53:55.500
Here what we will do is we need a formula that we could put countif and we could say in this range we could lock it down and it is not going to change for us.
00:53:55.500 --> 00:54:00.300
In this range countif is it is 63.
00:54:00.300 --> 00:54:07.300
Let us put all of that over however many different combinations there are.
00:54:07.300 --> 00:54:10.700
That is all of this.
00:54:10.700 --> 00:54:12.700
I think there is actually 20 combinations.
00:54:12.700 --> 00:54:18.000
In this way we can have Excel do the work for us.
00:54:18.000 --> 00:54:31.200
I am locking that down and once I do that it says that 2 out of 20 that ends up being 1 out of 10 which is .1.
00:54:31.200 --> 00:54:35.500
What we can do is just copy and paste this all the way down.
00:54:35.500 --> 00:54:51.200
We find that most of these are equally probable like 63, 65 minutes as the sum but 68 minutes is twice as likely because it is 4 out of 20.
00:54:51.200 --> 00:54:57.000
Twice as likely as this 70 minutes or 71 minutes.
00:54:57.000 --> 00:55:02.900
Here I will put probability of these particular sums.
00:55:02.900 --> 00:55:11.500
What ends up being helpful is to look back at the questions and say now that we have what we want to know here?
00:55:11.500 --> 00:55:18.300
If you randomly pick 2 shows to watch what is the probability that you will have finish watching both shows before your class.
00:55:18.300 --> 00:55:24.300
Let us say you just arrive at your class you could use all 74 minutes to watch.
00:55:24.300 --> 00:55:26.200
The cut off would be around here.
00:55:26.200 --> 00:55:33.800
We could watch all these different combinations and still be under 74 minutes.
00:55:33.800 --> 00:55:39.200
These are the only combinations that are not under 75 minutes.
00:55:39.200 --> 00:55:59.300
The way that we would write this in probable notation is p where x is the sum in this case is less than 74 minutes.
00:55:59.300 --> 00:56:05.200
You could just add these probabilities up using the addition rule.
00:56:05.200 --> 00:56:09.000
You could just add these probabilities up.
00:56:09.000 --> 00:56:22.000
In order to get that and we could put the sums and that should give us 80% because
00:56:22.000 --> 00:56:27.300
the only 2 we are leaving out are these and each of those have a probability of 10%.
00:56:27.300 --> 00:56:37.600
If you randomly pick shows to watch there is 80% probability that you will finish watching both of those shows before you go on to your class.
00:56:37.600 --> 00:56:43.400
Nice and helpful of probability to help this figure out our TV watching schedule.
00:56:43.400 --> 00:56:45.000
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