WEBVTT mathematics/probability/murray
00:00:00.000 --> 00:00:06.400
Hi and welcome back to the probability lectures here on www.educator.com, my name is Will Murray.
00:00:06.400 --> 00:00:12.100
We are starting a chapter now of working through our discrete distributions.
00:00:12.100 --> 00:00:15.800
The first we are going to study is the binomial distribution.
00:00:15.800 --> 00:00:19.700
This is also called the Bernoulli trials.
00:00:19.700 --> 00:00:26.200
If you are studying Bernoulli trials in your probability course then you are using the binomial distribution.
00:00:26.200 --> 00:00:29.200
These are synonymous terms for the same idea, let us learn what that idea is.
00:00:29.200 --> 00:00:37.000
Before I give you the formulas for the binomial distribution, I want to tell you the general setting.
00:00:37.000 --> 00:00:40.100
It is very important to be able to recognize this setting.
00:00:40.100 --> 00:00:45.700
When you get some random problem, you have to figure out is this a binomial distribution, is this a geometric distribution?
00:00:45.700 --> 00:00:49.600
Let me tell you the setting for the binomial distribution.
00:00:49.600 --> 00:00:55.900
It describes a sequence of N independent tests, each one of which can have 2 outcomes.
00:00:55.900 --> 00:01:01.100
You can think of running a test N times and each time you can either succeed or fail.
00:01:01.100 --> 00:01:06.000
Every time there is 2 outcomes, that is why it is called binomial, success or failure.
00:01:06.000 --> 00:01:09.200
It is also known as Bernoulli trials, as I mentioned.
00:01:09.200 --> 00:01:15.400
The technical example of the binomial distribution or a Bernoulli trial is flipping a coin.
00:01:15.400 --> 00:01:19.300
You want to think of flipping a coin exactly N times in a row.
00:01:19.300 --> 00:01:25.100
By the way, N is always constant for the binomial distribution or for Bernoulli trials.
00:01:25.100 --> 00:01:28.700
You always talk about N being a fixed number.
00:01:28.700 --> 00:01:34.200
That is different from some of the distributions we are going to encounter later like the geometric distribution.
00:01:34.200 --> 00:01:37.200
For the binomial distribution, N is always constant.
00:01:37.200 --> 00:01:43.000
We want to think about it, like I said, flipping a coin is the prototypical example.
00:01:43.000 --> 00:01:48.300
Although, that is somewhat limiting because people often think of the coin as being fair,
00:01:48.300 --> 00:01:53.100
meaning it got a 50-50 chance of coming up heads or tails.
00:01:53.100 --> 00:01:59.700
That certainly is a binomial distribution but you can also use a binomial distribution,
00:01:59.700 --> 00:02:02.700
even when the probabilities are not even like that.
00:02:02.700 --> 00:02:10.300
For example, if your coin is loaded, it is more likely to come up heads than tails, that is still a binomial distribution.
00:02:10.300 --> 00:02:13.300
We will see how we adjust the formulas to account for that.
00:02:13.300 --> 00:02:18.800
You can also think about any other kind of situation where you either have success or failure.
00:02:18.800 --> 00:02:27.300
For example, one sports team is going to play another sports team and each time your home team will either win or lose.
00:02:27.300 --> 00:02:29.200
That is a binomial distribution.
00:02:29.200 --> 00:02:35.400
If you say we are going to play the other team 15 × and each time we will win or lose.
00:02:35.400 --> 00:02:38.400
At the end, we will have a string of wins and losses.
00:02:38.400 --> 00:02:42.300
That is a binomial distribution, that is a set of Bernoulli trials.
00:02:42.300 --> 00:02:45.300
You can think of, for example, rolling a dice.
00:02:45.300 --> 00:02:50.800
You think of, wait a second, there are 6 different things that can happen when I roll a dice not just 2.
00:02:50.800 --> 00:02:56.500
Suppose you are only interested in whether the dice comes up 6 or not.
00:02:56.500 --> 00:03:00.200
If you roll a 6, it is a success and somebody pays you some money.
00:03:00.200 --> 00:03:02.500
If you do not roll a 6, it is a failure.
00:03:02.500 --> 00:03:12.800
1 through 5 essentially can sort of want those altogether and count those all as a single category of failure and rolling a 6 is a success.
00:03:12.800 --> 00:03:18.400
Essentially, rolling a dice just boils down to you roll it, do I get a 6, do I not get a 6?
00:03:18.400 --> 00:03:20.700
That is again, a binomial distribution.
00:03:20.700 --> 00:03:23.900
You can think of that as a set of Bernoulli trials.
00:03:23.900 --> 00:03:33.500
The probability of success if it is a fair dice, there is just 1/6 and the probability of failure is 5/6.
00:03:33.500 --> 00:03:39.700
There is all these different situations, they all come down to studying the binomial distribution.
00:03:39.700 --> 00:03:42.000
They already come down to the same formulas.
00:03:42.000 --> 00:03:47.300
Let us go ahead and take a look at the formulas that you get for the binomial distribution.
00:03:47.300 --> 00:03:52.600
You have several parameters, they go into this.
00:03:52.600 --> 00:03:55.400
You start out with the number of trials.
00:03:55.400 --> 00:03:58.800
I want to emphasize that that N is fixed.
00:03:58.800 --> 00:04:03.000
That is different from some of the other distributions that we are going to study later.
00:04:03.000 --> 00:04:11.300
In particular, the geometric distribution that we will study in the very next video, that it is different from this,
00:04:11.300 --> 00:04:18.400
and you keep flipping a coin until you get a head and that could take indefinitely long.
00:04:18.400 --> 00:04:20.700
The binomial distribution is not like that.
00:04:20.700 --> 00:04:25.300
You say ahead of time that I’m guaranteed I'm going to flip this coin N times,
00:04:25.300 --> 00:04:28.500
or we are going to play the other team N times.
00:04:28.500 --> 00:04:36.100
It is fixed, it stays constant throughout the experiment and you know that ahead of time.
00:04:36.100 --> 00:04:42.800
P is the probability of success on any given, looks like we got cutoff a little bit there.
00:04:42.800 --> 00:04:44.900
Let me just fill that in here.
00:04:44.900 --> 00:04:47.800
On any given trial,
00:04:47.800 --> 00:04:54.700
If you are dealing with a fair coin then P would be ½.
00:04:54.700 --> 00:05:02.100
If you are dealing with a sports team playing another sports team, it depends on the relative strength of the teams.
00:05:02.100 --> 00:05:10.800
Maybe if your team is the underdog, maybe they only win 1/3 of the games then P would be 1/3.
00:05:10.800 --> 00:05:15.700
That is your chance of winning any particular game with the other team.
00:05:15.700 --> 00:05:25.500
Maybe, if you are rolling a dice and you are trying to get a 6 then your probability of getting a 6 would be 1 and 6.
00:05:25.500 --> 00:05:34.300
Your probability of failure, now, we are going to call that Q but Q is not really very difficult to figure out
00:05:34.300 --> 00:05:41.200
because that is the probability of failing on any given trial, that is just 1 – P.
00:05:41.200 --> 00:05:48.800
The probability of failure is, we call it Q but sometimes we will swap back and forth and alternate between Q and 1 – P.
00:05:48.800 --> 00:05:50.900
They really mean the same thing.
00:05:50.900 --> 00:05:58.900
If you are flipping a coin and it is a fair coin then your probability of failing to get a head or getting a tail would be ½.
00:05:58.900 --> 00:06:08.100
If you have a sports team and we said the probability of winning any given match is 1/3 because we are the underdogs here,
00:06:08.100 --> 00:06:12.000
that means our probability of losing is 2/3.
00:06:12.000 --> 00:06:18.000
If we are rolling a dice and we are trying to get a 6, anything else is considered failure.
00:06:18.000 --> 00:06:24.400
The probability of getting anything other than a 6 is 5 out of 6 there.
00:06:24.400 --> 00:06:28.100
It is very easy to fill in Q, that is just 1 – P.
00:06:28.100 --> 00:06:35.200
The formula for probability distribution, there are some terrible notation here and I do not like to use it
00:06:35.200 --> 00:06:39.100
but it is kind of universal in the probability textbooks.
00:06:39.100 --> 00:06:41.200
We are forced to deal with it.
00:06:41.200 --> 00:06:47.200
This P of Y here, what that represents is the probability of Y successes.
00:06:47.200 --> 00:06:49.000
Let me write that in.
00:06:49.000 --> 00:07:04.600
That is the probability of exactly Y successes.
00:07:04.600 --> 00:07:10.600
If you are flipping a coin N ×, this is the probability that you will get exactly Y heads.
00:07:10.600 --> 00:07:19.600
If your sports team is going to play the other team N ×, this is the probability that you will win exactly Y games.
00:07:19.600 --> 00:07:27.600
If you are going to roll a dice N ×, this is the probability that you will get exactly Y successes.
00:07:27.600 --> 00:07:32.300
The formula that we have here is N choose Y, that is not a fraction there.
00:07:32.300 --> 00:07:34.300
That is really N choose Y.
00:07:34.300 --> 00:07:40.500
The other notation that we have for that is the binomial coefficient notation C of NY.
00:07:40.500 --> 00:07:50.600
The actual way you calculate that is as N! ÷ Y! × N – Y!.
00:07:50.600 --> 00:07:57.300
It is not just a fraction N/Y, that is really the formula for combinations.
00:07:57.300 --> 00:08:02.800
N!/Y! × N – Y!.
00:08:02.800 --> 00:08:08.300
The rest of this formula here, we have P ⁺Y and here is the really unpleasant part here.
00:08:08.300 --> 00:08:13.200
This P right here is not the same as the P on the left hand side.
00:08:13.200 --> 00:08:19.100
I said the P on the left hand side is the probability of exactly Y successes.
00:08:19.100 --> 00:08:30.600
This P right here is this P right here, it is the probability of getting a success on any given trial.
00:08:30.600 --> 00:08:38.400
It is this P up here, whatever the probability of successes on any given trial, that is what you fill in for this P.
00:08:38.400 --> 00:08:42.800
That is the really unfortunate notation that you see with the binomial distribution,
00:08:42.800 --> 00:08:47.600
is that they use P for 2 different things in the same formula.
00:08:47.600 --> 00:08:56.100
I think that is really a high crime there but I was not given the choice to make up the notation myself.
00:08:56.100 --> 00:09:01.200
I do not want to mislead you by using different notation from everybody else in the world.
00:09:01.200 --> 00:09:03.000
We are kind of stuck with that.
00:09:03.000 --> 00:09:08.500
Just be careful there that that P and that P are 2 different uses of the word P.
00:09:08.500 --> 00:09:12.200
The problem with probability is everything starts with P.
00:09:12.200 --> 00:09:16.400
We end up using the variable P in many places.
00:09:16.400 --> 00:09:19.600
We have the exponent Y, we have Q.
00:09:19.600 --> 00:09:24.300
Remember, we said Q is just 1 – P, that is easy to fill in and N – Y,
00:09:24.300 --> 00:09:28.700
that is our formula, and then let us think about the range of values that Y could be.
00:09:28.700 --> 00:09:33.600
If you are going to flip a coin N ×, how many heads could you get?
00:09:33.600 --> 00:09:37.000
The fewest heads you can get would be 0, if you do not get any heads at all.
00:09:37.000 --> 00:09:40.000
The most heads you can get would be N heads.
00:09:40.000 --> 00:09:45.300
Our range of possibilities for Y is from 0 to N.
00:09:45.300 --> 00:09:49.000
That is a formula we will be using over and over again.
00:09:49.000 --> 00:09:54.500
There are another couple of issues we need to straighten out, before we jump into the examples.
00:09:54.500 --> 00:09:56.300
Let us take a look at those.
00:09:56.300 --> 00:10:02.800
The key properties of a binomial distribution and we will need to know these properties for every distribution we encounter.
00:10:02.800 --> 00:10:13.100
The binomial distribution is just the first one, but we will be getting into the geometric distribution and the Poisson distribution,
00:10:13.100 --> 00:10:15.000
all these other distributions later on.
00:10:15.000 --> 00:10:19.300
For every single one, we want to know the mean, variance, and the standard deviation.
00:10:19.300 --> 00:10:21.800
Here they are for the binomial distribution.
00:10:21.800 --> 00:10:29.200
The mean, which is also known as the expected value means the exact same thing, mean and expected value.
00:10:29.200 --> 00:10:33.800
There are 2 different notations for it.
00:10:33.800 --> 00:10:40.900
We use the Greek letter μ for mean and we also say E of Y for expected value.
00:10:40.900 --> 00:10:42.800
Those are really the same thing.
00:10:42.800 --> 00:10:44.600
People also some× say the average.
00:10:44.600 --> 00:10:49.000
All those things are essentially should be used synonymously.
00:10:49.000 --> 00:10:51.300
But it is N × P.
00:10:51.300 --> 00:10:56.800
Remember, N is the number of trials and P is the probability of success on any given trial.
00:10:56.800 --> 00:11:04.800
The variance, two different notations for that, V of Y and σ² is N × P × Q.
00:11:04.800 --> 00:11:08.500
Remember, Q is just 1 – P.
00:11:08.500 --> 00:11:13.600
Sometimes you will see that written as N × P × 1 – P but they mean the same thing.
00:11:13.600 --> 00:11:17.900
Standard deviation is always just the square root of the variance.
00:11:17.900 --> 00:11:24.800
If you tell me the variance, I can always calculate the standard deviation very easily.
00:11:24.800 --> 00:11:31.000
Just take the square root of the variance and that is the square root of N × P × Q.
00:11:31.000 --> 00:11:34.300
We will be calculating those in some of our examples.
00:11:34.300 --> 00:11:38.200
Let us go ahead and get started with those.
00:11:38.200 --> 00:11:46.000
In our first example, we have the Los Angeles angels are going to play the Tasmanian devils in a 5 game series.
00:11:46.000 --> 00:11:50.800
Maybe this is football or baseball, let us say baseball.
00:11:50.800 --> 00:11:54.500
The angels have a 1/3 chance of winning any given game.
00:11:54.500 --> 00:11:57.700
I guess the Tasmanian devils are bit stronger than the Angels.
00:11:57.700 --> 00:12:03.700
What is the chance that the Angels will win exactly 3 games here?
00:12:03.700 --> 00:12:09.700
Let me write down our general formula for the binomial distribution.
00:12:09.700 --> 00:12:21.600
The general formula for binomial distribution is P of Y is equal to N choose Y, binomial coefficient there, combinations N choose Y.
00:12:21.600 --> 00:12:26.900
P ⁺Y × Q ⁺N – Y.
00:12:26.900 --> 00:12:30.100
Let me fill in everything I know here.
00:12:30.100 --> 00:12:36.800
The Y I’m interested in is 3 games because I want to find the chance that they are going to win exactly 3 games.
00:12:36.800 --> 00:12:49.400
My Y will be 3, my P is the probability that the Angels will win any particular game and that is 1/3 there.
00:12:49.400 --> 00:12:53.100
My N is the number of games that they are going to play in total.
00:12:53.100 --> 00:13:02.300
It is a 5 game series, that is where I’m getting that from, N = 5 and Q is just 1 – P.
00:13:02.300 --> 00:13:08.000
Q is 1 - P which is 2/3, 1 -1/3 there.
00:13:08.000 --> 00:13:32.100
The probability of 3, probability of winning 3 games is 5 choose 3 N choose Y 5 choose 3 × 1/3³ × 2/3⁵ – Y.
00:13:32.100 --> 00:13:40.400
Be careful here, it is rather seductive to get your binomial coefficients and your fractions mix up here
00:13:40.400 --> 00:13:43.600
because we are mixing them both on the same formula.
00:13:43.600 --> 00:13:49.800
The fractions are 1/3 and 2/3, that 5/3 is a binomial coefficient, it is not a fraction.
00:13:49.800 --> 00:14:03.100
We want to expand that out as a binomial coefficient, 5!/3! × 2! × 1/3.
00:14:03.100 --> 00:14:16.200
I got 1/3³ and then 1/3³ × 2/3⁵ – 3, that is 2/3².
00:14:16.200 --> 00:14:21.600
5!/3! that means the 1, 2, 3, will cancel out.
00:14:21.600 --> 00:14:25.700
We would just get 5 × 4/2 ×,
00:14:25.700 --> 00:14:34.000
Let us see, I’m going to have 3²/5 in the denominator here because I got 3 copies of 1/3
00:14:34.000 --> 00:14:40.200
and then 2 more here and 2² in the numerator.
00:14:40.200 --> 00:14:50.700
5 × 4/2 that is 20/2 is 10 × 2²/3⁵.
00:14:50.700 --> 00:14:56.500
10 × 2², 2² is 4, 10 × 4 is 40.
00:14:56.500 --> 00:15:18.500
3⁵, 3 × 3 is 9 × 3 is 27 × 3 is 81 × 3 is 243.
00:15:18.500 --> 00:15:23.400
That is my exact probability of winning exactly 3 games.
00:15:23.400 --> 00:15:28.700
443 is just about 6 × 40 because 240 is 6 × 40.
00:15:28.700 --> 00:15:34.600
This is very close to 1/6, if you want to make an approximation there.
00:15:34.600 --> 00:15:43.800
The exact value would be 40/243.
00:15:43.800 --> 00:15:48.200
The wraps that one up, let us just see how we solve that.
00:15:48.200 --> 00:15:51.900
I used my generic formula for the binomial distribution.
00:15:51.900 --> 00:16:01.100
The probability of exactly Y successes is N choose Y × P ⁺Y × Q ⁺N – Y.
00:16:01.100 --> 00:16:03.500
I’m going to fill in all the numbers that I know here.
00:16:03.500 --> 00:16:07.400
My N came from the fact that it was a 5 game series.
00:16:07.400 --> 00:16:14.000
My Y is the number of game that I want to win, that was the 3 here.
00:16:14.000 --> 00:16:22.800
The 1/3 is the little P, that is the probability that I will win any particular game.
00:16:22.800 --> 00:16:26.200
My Q is just 1 – P, that is 2/3.
00:16:26.200 --> 00:16:32.200
I drop all those numbers in here, very careful, the 5 choose 3 is a binomial coefficient.
00:16:32.200 --> 00:16:39.500
It is a combination, it is not a fraction.
00:16:39.500 --> 00:16:46.900
I simplify these fractions down while I’m simplifying down the binomial coefficient there.
00:16:46.900 --> 00:16:54.500
Doing the arithmetic, it simplifies down to 40/243 which I noticed is approximately equal to 1/6.
00:16:54.500 --> 00:17:02.800
That is my probability of winning exactly 3 games out of this 5 game series.
00:17:02.800 --> 00:17:11.000
In example 2 here, we got a big exam coming up and we studied most of the material but not all of it.
00:17:11.000 --> 00:17:15.900
In fact any given problem, we have a ¾ chance of getting that problem right.
00:17:15.900 --> 00:17:19.100
Most likely, we will get a problem right but not guaranteed.
00:17:19.100 --> 00:17:28.700
The exam that we are going to take is 10 problems and I guess we are really hoping to get an 80% score or better.
00:17:28.700 --> 00:17:35.300
I would like to score 80% on this exam but I really only studied ¾ of the problems.
00:17:35.300 --> 00:17:39.700
This is really a binomial distribution problem.
00:17:39.700 --> 00:17:49.100
Remember that you use the binomial distribution, when you have a sequence of trials and each trial ends in success or failure.
00:17:49.100 --> 00:17:51.200
How does this correspond to that?
00:17:51.200 --> 00:17:57.600
We have 10 problems here, each problem we will do our best to solve it and will either succeed or fail.
00:17:57.600 --> 00:18:01.300
It is exactly 10 problems, each one is N success or failure.
00:18:01.300 --> 00:18:04.700
That is definitely a binomial distribution.
00:18:04.700 --> 00:18:08.100
Let me go ahead and set up the generic formula for binomial distribution.
00:18:08.100 --> 00:18:21.800
The probability of getting exactly Y successes is N choose Y × P ⁺Y × Q ⁺N – Y.
00:18:21.800 --> 00:18:26.800
In this case, let me fill in here, while my N is the number of trials here.
00:18:26.800 --> 00:18:33.700
That is the number of problems I will be struggling with, N is 10.
00:18:33.700 --> 00:18:40.900
P is my probability of getting any particular problem right.
00:18:40.900 --> 00:18:44.300
We were told that that is ¾.
00:18:44.300 --> 00:18:49.100
My Y is the number of successes that I would like to have.
00:18:49.100 --> 00:18:57.400
In this case, I want to score 80% or better which means out of 10 problems, I got to get 8 of them, or 9 of them, or 10 of them right.
00:18:57.400 --> 00:19:03.600
Our Y, in turn be 8, 9, and 10.
00:19:03.600 --> 00:19:08.000
We have several calculations here.
00:19:08.000 --> 00:19:15.100
The Q, remember is always 1 – P, that is our chance of failure on any given problem.
00:19:15.100 --> 00:19:26.400
1 – P, 1 -3/4 is ¼, if you give me a single problem, that is the chance I will not get it right.
00:19:26.400 --> 00:19:33.000
Since, I need to find the probability of getting exactly 8, 9, or exactly 10 problems.
00:19:33.000 --> 00:19:37.600
I will be adding up 3 different quantities here, P of 8.
00:19:37.600 --> 00:19:40.700
I will give myself some space because I do not get a little bit messy.
00:19:40.700 --> 00:19:48.300
+ P of 9 + P of getting exactly 10 problems right.
00:19:48.300 --> 00:19:50.400
I will workout each one of those.
00:19:50.400 --> 00:20:00.300
P of 8, just dropping Y=8 into this formula, is 10 choose 8, N was 10 × P ⁺Y.
00:20:00.300 --> 00:20:08.100
P is ¾ ⁺Y is 8 × Q was ¼ ⁺N- Y.
00:20:08.100 --> 00:20:39.100
N was 10 so N – 8 is 2 + P of 9 that is 10 choose 9 × ¾⁹ × ¼¹ + P of 10 is 10 choose 10 × 3/3 ⁺10 × ¼ ⁺10 – 10 which is 0.
00:20:39.100 --> 00:20:41.700
I want to simplify that, these numbers are going to get a bit messy.
00:20:41.700 --> 00:20:45.800
At some point, I’m going to throw out my hands in despair and just go to the calculator.
00:20:45.800 --> 00:20:50.400
Let me simplify a bit on paper first.
00:20:50.400 --> 00:20:53.700
In particular, these binomial coefficients, I know how to simplify those.
00:20:53.700 --> 00:20:56.900
Remember, you are not going to mix up the fractions.
00:20:56.900 --> 00:21:07.000
This 10 choose 8, that is 10!/8!/10 -8! = 2!.
00:21:07.000 --> 00:21:13.500
The 10! And 8! Cancel each other just leaving 10 × 9/2.
00:21:13.500 --> 00:21:33.200
That is 45 there, 45 × 3⁸/4 ⁺10 because we have 8 factors of 3 and 8 factors of 4 and then 2 more factors of 4.
00:21:33.200 --> 00:21:41.000
10 choose 9 here is 10!/9! × 1!.
00:21:41.000 --> 00:21:50.200
That is just 10!/9! which is all the factors are cancel except the 10 there.
00:21:50.200 --> 00:21:59.400
10 × 3⁹/4 ⁺10.
00:21:59.400 --> 00:22:03.100
Finally, we had 10 choose 10.
00:22:03.100 --> 00:22:07.200
There is only one way to choose 10 things out of 10 possibilities.
00:22:07.200 --> 00:22:15.500
In case you want to confirm that to the formula, it is 10! ÷ 10! × 10 - 10 is 0!.
00:22:15.500 --> 00:22:21.200
But 0! Is just 1, remember, so that is 1.
00:22:21.200 --> 00:22:31.300
That is 1 × ¾ ⁺10, 3 ⁺10/4 ⁺10, that ¼⁰ is just 1, that does not do anything.
00:22:31.300 --> 00:22:39.600
At this point, I do not think the numbers are going to get any nicer by trying to simplify them as fractions.
00:22:39.600 --> 00:22:42.900
I’m going to go ahead and threw these numbers in, all these numbers to my calculator.
00:22:42.900 --> 00:22:45.100
Let me show you what I got for each one of those.
00:22:45.100 --> 00:23:02.800
For the first one, I got the 0.2816 +, in the second one I got 0.1877 +, in the last part I got 0.0563.
00:23:02.800 --> 00:23:08.600
What these really represent these 3 numbers right now, represent your probabilities
00:23:08.600 --> 00:23:14.100
of getting exactly 8 problems right, that is P of 8 right there.
00:23:14.100 --> 00:23:19.100
The probability that you get exactly 8 problems right, you score exactly 80% on the test.
00:23:19.100 --> 00:23:25.100
This is probability of getting 90% on the test so you got exactly 9 problems right.
00:23:25.100 --> 00:23:28.400
This is your probability of getting 100%, getting all 10 problems right.
00:23:28.400 --> 00:23:38.700
Not very likely, you got 5% chance every single test, if you are only ready with ¾ of the material going in.
00:23:38.700 --> 00:23:53.400
If we add those up, 28 + 18 + 5 turns out to be, I did this on my calculator, 52.56 is approximately,
00:23:53.400 --> 00:24:05.200
52.5%, I will round that up to 53%, that is your probability of getting 80% or more on this exam.
00:24:05.200 --> 00:24:13.400
You studied ¾ of the material, your probability of getting 80% on the exam is 53%.
00:24:13.400 --> 00:24:16.200
Let me show you how I derived that.
00:24:16.200 --> 00:24:20.800
I started with the basic formula for the binomial distribution, here it is.
00:24:20.800 --> 00:24:24.500
And then, I filled in all the quantities I know.
00:24:24.500 --> 00:24:28.800
The N = 10 that come from the stem of the problem.
00:24:28.800 --> 00:24:40.300
The P, the probability of getting any problem right is ¾, that also comes from the stem of the problem.
00:24:40.300 --> 00:24:48.400
The Y that we are interested in, we want to get 80% or better, that means that we want to get 8 problems,
00:24:48.400 --> 00:24:50.900
or 9 problems, or 10 problems right.
00:24:50.900 --> 00:24:57.600
Because if you are shooting for 80% and if you end up getting 90 or 100, that certainly is acceptable.
00:24:57.600 --> 00:25:00.300
We have to add up all those different possibilities.
00:25:00.300 --> 00:25:05.900
The Q is always 1 – P, since P was ¾, Q was ¼.
00:25:05.900 --> 00:25:12.100
We just drop those in for the different values of Y, the 8, 9, and 10.
00:25:12.100 --> 00:25:20.300
We get these binomial coefficients and some fairly nasty fractions which I did not want to simplify by hand.
00:25:20.300 --> 00:25:25.400
We sorted out the binomial coefficients into 45 and 10 and 1.
00:25:25.400 --> 00:25:29.800
Each one of those multiplied by some fractions gave me some percentages,
00:25:29.800 --> 00:25:33.900
the probabilities of getting 8 problems, 9 problems, 10 problems right.
00:25:33.900 --> 00:25:40.600
We would put those all together and we get a total probability of 53%.
00:25:40.600 --> 00:25:52.300
If you are shooting for an 80% on an exam and you study 3/4 of the material, your chance of getting that 80% is 53%.
00:25:52.300 --> 00:25:56.400
You are likely get 80% but it is definitely not a sure thing.
00:25:56.400 --> 00:26:01.000
You might want to study a little more there.
00:26:01.000 --> 00:26:05.400
Example 3, we are going to keep going with that same exam from example 2.
00:26:05.400 --> 00:26:13.000
It is telling us that each problem is worth 10 points, what is your expected grade and your standard deviation?
00:26:13.000 --> 00:26:15.600
Remember, expected grade is a technical term.
00:26:15.600 --> 00:26:26.200
As soon as you see the word expected in a probability problem, that does not mean the English meaning of the word expected.
00:26:26.200 --> 00:26:35.800
That does not mean what grade you are going to get but that means on average, what your grade be if you take this exam many times.
00:26:35.800 --> 00:26:39.000
What is your average going to be?
00:26:39.000 --> 00:26:45.000
It is asking for the expected value of your exam score.
00:26:45.000 --> 00:26:49.800
We have learned the formula for the expected value of the binomial distribution.
00:26:49.800 --> 00:26:52.800
That is the same as the mean of the binomial distribution.
00:26:52.800 --> 00:26:59.700
The formula we learned this back on the third slide of this lecture, it is N × P.
00:26:59.700 --> 00:27:09.100
In this case, the N was 10 and the P is the probability of you getting any particular problem right, that was ¾.
00:27:09.100 --> 00:27:18.900
The expected value of Y, I’m using Y here to mean the number of problems that you get right.
00:27:18.900 --> 00:27:22.300
I should probably clarify that a little earlier.
00:27:22.300 --> 00:27:30.800
Problems you get right.
00:27:30.800 --> 00:27:38.400
We just figured out down below that that is 10 × ¾ is 7 ½.
00:27:38.400 --> 00:27:47.100
That 7 ½ problems but each problem is 10 points.
00:27:47.100 --> 00:27:55.400
That is 75 points on the exam, that is your expecting great.
00:27:55.400 --> 00:28:01.100
Remember, I said that that is a technical term, that is you are expected grade.
00:28:01.100 --> 00:28:09.700
In real life, there is no way you can get a 75 on the exam because all the problems are worth 10 points each.
00:28:09.700 --> 00:28:19.100
In real life, when you take a single exam, you will have to get a multiple of 10.
00:28:19.100 --> 00:28:27.600
You might get a 60% on the exam, a 70, 80, etc.
00:28:27.600 --> 00:28:35.400
You will not get a 75 on the exam, I guarantee you because we are not talking about partial credit here.
00:28:35.400 --> 00:28:41.400
Your actual score would be 60, 70, 80, or so on.
00:28:41.400 --> 00:28:48.900
What I mean when I say that your expected grade is 75, what I mean is that if you take this exam many times,
00:28:48.900 --> 00:29:12.600
or if you take many exams, your average over the long run will be 75 points.
00:29:12.600 --> 00:29:21.100
Maybe, for example if you take 2 exams, your total on the 2 exams might be 150 which means you are averaging 75 points per exam,
00:29:21.100 --> 00:29:26.200
even though you are not going to get exactly 75 points on any exams here.
00:29:26.200 --> 00:29:34.200
That was your expecting grade, your standard deviation, a good steppingstone to calculating that is to find the variance first.
00:29:34.200 --> 00:29:38.100
Let us find the variance of Y, variance of your score.
00:29:38.100 --> 00:29:46.200
Variance, we learned the formula for that, it was also on the third slide of this video, NPQ.
00:29:46.200 --> 00:29:53.100
Our N here is 10, our P is ¾, and our Q is ¼.
00:29:53.100 --> 00:29:56.600
Remember, Q is always 1 – P.
00:29:56.600 --> 00:30:04.800
If we simplify that, we get 30/16 that is not extremely revealing at this point.
00:30:04.800 --> 00:30:08.300
But remember, that was just the variance, that is not our standard deviation.
00:30:08.300 --> 00:30:13.600
To get the standard deviation, you take the square root of the variance.
00:30:13.600 --> 00:30:19.600
Our σ is the square root of the variance, that is always true.
00:30:19.600 --> 00:30:33.300
It is √ 38/16 and I can simplify that a bit into √30 on top and √16 is just 4.
00:30:33.300 --> 00:30:39.300
It does not really do anything good after that, I just threw it into a calculator.
00:30:39.300 --> 00:30:48.300
What it came back with was that that is approximately equal to 1.369 problems.
00:30:48.300 --> 00:30:57.000
Our unit here is the problem because Y was measured in the number of problems that we get right, 1.369 problems.
00:30:57.000 --> 00:31:15.900
Our standard deviation in terms of points on the exam, that is equal to 13.69 points because each problem was worth 10 points.
00:31:15.900 --> 00:31:30.600
Our σ there is 13, it is approximately equal to 13.69 points on the exam.
00:31:30.600 --> 00:31:36.300
You can estimate your score on the exam, your expected grade would be 75 points.
00:31:36.300 --> 00:31:46.600
Your standard deviation as you take many exams will be 13.69 points up above and below 75 points.
00:31:46.600 --> 00:31:48.700
Let me recap how we calculated that.
00:31:48.700 --> 00:31:55.600
This really came back to remembering the formulas from the third slide that we had earlier on the videos.
00:31:55.600 --> 00:32:00.700
If you do not remember those, just go back, check them out on the third slide of this video and you will see them.
00:32:00.700 --> 00:32:07.300
The expected value is NP, the variance is NPQ.
00:32:07.300 --> 00:32:11.000
Now, I’m just dropping in our values for N.
00:32:11.000 --> 00:32:18.800
N is 10, P is ¾, our Q is 1 - P is ¼.
00:32:18.800 --> 00:32:25.900
That tells us the expected value and the variance, in terms of the problems on the exam
00:32:25.900 --> 00:32:32.800
because we define our random variable in terms of the problems that we expect to get right.
00:32:32.800 --> 00:32:39.700
To convert into actual points on the exam, we multiplied by 10 because each problem is worth 10 points.
00:32:39.700 --> 00:32:48.400
7 1/2 problems I expect to get 7 1/2 problems right that means I expect on average to get 75 points on the exam.
00:32:48.400 --> 00:32:56.500
I will never get exactly 75 because with 10 point problems, my score will definitely be some multiple of 10.
00:32:56.500 --> 00:33:01.100
But on average, if I take many exams, I will get 75 points.
00:33:01.100 --> 00:33:07.800
The variance, drop in those numbers I get 30/16, that is the variance not the standard deviation.
00:33:07.800 --> 00:33:16.000
To get the standard deviation, you take the √ of that and that simplifies down into 1.369 problems.
00:33:16.000 --> 00:33:28.200
Converting that to points gives me a standard deviation of 13.69 points on this exam.
00:33:28.200 --> 00:33:35.300
In example 4, we are going watch the heralded Long Beach jack rabbits play and
00:33:35.300 --> 00:33:38.800
they are going to be playing in the world pogo sticking championship.
00:33:38.800 --> 00:33:45.000
Apparently, they are very good at this, as you expect jack rabbits to be.
00:33:45.000 --> 00:33:48.900
Each year they have an 80% chance of winning the world championships.
00:33:48.900 --> 00:33:54.900
They are obviously the dominant force in the world pogo sticking championships.
00:33:54.900 --> 00:34:02.200
The question is we want to find the probability that they will win exactly 5 × in the next 7 years.
00:34:02.200 --> 00:34:08.400
7 years of championship, they will play every year, each year they got 80% chance.
00:34:08.400 --> 00:34:16.200
We also want to find the probability that they will win at least 5 × in the next 7 years.
00:34:16.200 --> 00:34:17.500
We are going to calculate both those.
00:34:17.500 --> 00:34:19.500
We need a little more space for this.
00:34:19.500 --> 00:34:24.100
I put an extra slide in here for us to work these out.
00:34:24.100 --> 00:34:31.300
This is still the example of the long beach poly jack rabbits in their pogo sticking championship.
00:34:31.300 --> 00:34:36.800
We are going to be playing 7 championships here.
00:34:36.800 --> 00:34:40.400
Again, this is really a Bernoulli trial.
00:34:40.400 --> 00:34:44.200
Why is this a Bernoulli trial, it is because we are playing 7 championships.
00:34:44.200 --> 00:34:47.400
Each year we will win or we will lose.
00:34:47.400 --> 00:34:50.800
We have a probability of winning each year or losing each year.
00:34:50.800 --> 00:34:55.200
Let me fill in the generic formula for Bernoulli trials.
00:34:55.200 --> 00:35:09.900
P of getting exactly Y successes is N choose Y × P ⁺Y × Q ⁺N – Y.
00:35:09.900 --> 00:35:15.700
That is our generic binomial distribution formula.
00:35:15.700 --> 00:35:18.000
Let me fill in whatever values I can.
00:35:18.000 --> 00:35:22.600
The N here is the number of trials that we are doing here.
00:35:22.600 --> 00:35:28.800
We are going to track this over 7 years so our N is 7.
00:35:28.800 --> 00:35:34.000
P is our probability of success on any given trial.
00:35:34.000 --> 00:35:38.000
That is the probability that the jack rabbits will win in any given year.
00:35:38.000 --> 00:35:39.100
We said that that is 80%.
00:35:39.100 --> 00:35:46.700
I will give that as a fraction, I will try to work this not using fractions, that is 4/5.
00:35:46.700 --> 00:35:55.200
Q is always 1 – P so that is 1 - 4/5.
00:35:55.200 --> 00:35:57.500
In this case, that is 1/5.
00:35:57.500 --> 00:36:02.300
Finally, what is the Y value that we will be interested in here?
00:36:02.300 --> 00:36:09.000
Our Y value, we want to win exactly 5 × for the first part of the problem.
00:36:09.000 --> 00:36:13.200
We want to find the probability of exactly Y successes.
00:36:13.200 --> 00:36:16.600
In the second part of the problem, we want to win at least 5 ×.
00:36:16.600 --> 00:36:22.300
That means we could win 5 ×, we could win 6 ×, we could win 7 ×.
00:36:22.300 --> 00:36:25.300
Let us calculate all of those.
00:36:25.300 --> 00:36:56.200
P of 5, when I plug in Y = 5 here, that is 7 choose 5 × P is 4/5 ⁺Y is 5 × Q is 1/5 ⁺N – Y, that 7 – 5 is 2.
00:36:56.200 --> 00:37:00.000
Now, I just have to expand and simplify these fractions.
00:37:00.000 --> 00:37:04.200
7 choose 5 that is not 7 ÷ 5.
00:37:04.200 --> 00:37:14.000
7 choose 5 is 7!/5! × 2!.
00:37:14.000 --> 00:37:16.100
Let me cancel the factorials.
00:37:16.100 --> 00:37:27.100
That is just 7 × 6 because the 5! takes care of all the other factors, ÷ 2!.
00:37:27.100 --> 00:37:35.200
That is equal to, 7 × 6, 6/2 is 3, 7 × 3 is 21.
00:37:35.200 --> 00:37:54.500
I still have 4⁵/5⁷ because there are five 5 in the first fraction and two 5 in the second fraction.
00:37:54.500 --> 00:38:07.100
What will we get here is 21 × 4⁵ ÷ 5⁷.
00:38:07.100 --> 00:38:11.500
That does not really turn out to be any particularly interesting numbers.
00:38:11.500 --> 00:38:18.100
I just left that as a fraction, I did not bother to plug that into my calculator but that is our answer to part A.
00:38:18.100 --> 00:38:25.700
That is the probability that the jack rabbits will come home with exactly 5 championships within the next 7 years.
00:38:25.700 --> 00:38:30.800
In part B, we want to get at least 5 championships.
00:38:30.800 --> 00:38:39.300
That means we really want to figure out the probability of getting 5 or 6, or7 championships.
00:38:39.300 --> 00:38:44.100
We figured out the probability of 5 already, let us find the probability of 6.
00:38:44.100 --> 00:39:02.000
We use a same formula except we put in Y equal 6, so 7 choose 6, 4/5⁶, 1/5⁷ -6 that is 1 there.
00:39:02.000 --> 00:39:09.200
7 choose 6 is 7!/6! × 1!.
00:39:09.200 --> 00:39:18.600
And then we have of 4⁶/5⁶ × 5 ⁺15⁷.
00:39:18.600 --> 00:39:28.500
7!/6! Is 7, that 7 × 4⁶ or 5⁷.
00:39:28.500 --> 00:39:31.700
Not a particularly interesting number by itself.
00:39:31.700 --> 00:39:37.200
Let me go ahead and figure out P of 7, the probability of winning all 7 matches.
00:39:37.200 --> 00:39:42.000
I’m going to use the binomial distribution formula.
00:39:42.000 --> 00:39:46.500
Although, it might be a little easier if you think about this directly but I want to practice the formula.
00:39:46.500 --> 00:40:01.200
It is 7 choose 7 × 4/5⁷ × 1/5⁷ -7 which is 0.
00:40:01.200 --> 00:40:07.600
7 choose 7 is 7!/7! × 0!.
00:40:07.600 --> 00:40:15.000
We just have a 4/5⁷ because the 1/5⁰ is just 1.
00:40:15.000 --> 00:40:20.700
7!/7! Is just 1, we get 4⁷/5⁷.
00:40:20.700 --> 00:40:28.500
It would have been easier to think about that as saying we have a 4/4 chance of winning.
00:40:28.500 --> 00:40:33.100
In order to win 7 matches, we have to win all 7 × in a row.
00:40:33.100 --> 00:40:38.400
It is 4/5 × 4/4, 4/5⁷.
00:40:38.400 --> 00:40:47.400
That is probably an easier way to get there more directly but I just want to practice using the probability distribution formula.
00:40:47.400 --> 00:40:55.200
The probability that we will win at least 5 matches, you just add up those 3 numbers.
00:40:55.200 --> 00:41:01.900
The probability of 5 + the probability of 6 + the probability of 7.
00:41:01.900 --> 00:41:06.000
The fractions are actually fun to workout here, I did work them out.
00:41:06.000 --> 00:41:13.400
I will work them out, 21 × 4⁵/5⁷.
00:41:13.400 --> 00:41:20.500
P of 6 is 7 × 4⁶/5⁷.
00:41:20.500 --> 00:41:25.300
P of 7 is just 4⁷/5⁷.
00:41:25.300 --> 00:41:28.100
We can factor out that 5⁷ in the denominator.
00:41:28.100 --> 00:41:36.500
Also, in the numerator we got lots of 4 and we factor out 4⁵/5⁷.
00:41:36.500 --> 00:41:49.800
The numerator, I still have a 21 + 7 × 4 because 4⁶ is 4⁵ × 4 + 4².
00:41:49.800 --> 00:41:56.500
7 × 4 is 28, 4² is 16.
00:41:56.500 --> 00:42:01.000
This is 4⁵/5⁷.
00:42:01.000 --> 00:42:10.500
21 + 28 is 49 + 16 is 65.
00:42:10.500 --> 00:42:26.100
But 65 is 5 × 13, we cancel out one of those 5’s and we get 4⁵ × 13/5⁶
00:42:26.100 --> 00:42:30.700
because one of the 5 was canceled with the 5 from the 65.
00:42:30.700 --> 00:42:36.000
We divide from the numerator there.
00:42:36.000 --> 00:42:45.200
That is our probability of winning at least 5 games or 5 championships over the next 7 years.
00:42:45.200 --> 00:42:47.100
Let me recap here.
00:42:47.100 --> 00:42:52.100
This is kind of a classic binomial distribution problem, classic Bernoulli trials.
00:42:52.100 --> 00:42:56.500
It does not have to be coin flipping even though people always talk about coin flipping.
00:42:56.500 --> 00:43:02.400
In this case, it is a question of the Long Beach jack rabbits in the pogo sticking championships.
00:43:02.400 --> 00:43:06.100
Every year, they either win or they lose.
00:43:06.100 --> 00:43:10.000
They win with a probability of 4/5.
00:43:10.000 --> 00:43:13.300
They lose with a probability of 1/5.
00:43:13.300 --> 00:43:19.000
They will play for 7 years, that is why we have our N = 7.
00:43:19.000 --> 00:43:26.800
The question was asking us what our chance of winning exactly 5 × is, that is where we get that 5.
00:43:26.800 --> 00:43:30.900
And then later on, we want the probability of winning at least 5 ×.
00:43:30.900 --> 00:43:33.700
We are going to calculate 6 and 7 as well.
00:43:33.700 --> 00:43:43.400
What we are really doing here is dropping 5, and 6, and 7 in for Y into the binomial distribution formula.
00:43:43.400 --> 00:43:58.100
I just dropped Y = 5, that 5 in the denominator came from the probability but that came from Y there and that 2 was 7 – Y.
00:43:58.100 --> 00:44:06.600
And then, I simplified all the fractions and I got that is my probability of winning exactly 5 championships.
00:44:06.600 --> 00:44:15.500
I found the probability of winning 6 championships the same way by running all the way through Y = 6, then I ran through Y=7.
00:44:15.500 --> 00:44:22.700
There is an easier way to calculate that but I want to practice the binomial distribution formula.
00:44:22.700 --> 00:44:29.200
Add all those together to find the probability of winning at least 5 × because it is at least 5 ×,
00:44:29.200 --> 00:44:31.700
that is why you put a greater than or equal to.
00:44:31.700 --> 00:44:35.400
We got to check 5, 6, and 7.
00:44:35.400 --> 00:44:39.300
We have a common denominator 5⁷ on all of these.
00:44:39.300 --> 00:44:46.000
These numbers, you can do some nice factoring, factor out 4⁵ and simplify everything down
00:44:46.000 --> 00:44:56.100
and you get still bit of a cumbersome number but that tell us our probability of winning at least 5 × in 7 years.
00:44:56.100 --> 00:44:59.600
I want you to hang onto the numbers from this example because in example 5,
00:44:59.600 --> 00:45:04.400
which we are about to do refers back to this example.
00:45:04.400 --> 00:45:11.300
It is the same scenario with the long beach poly jack rabbits playing in the pogo sticking championship.
00:45:11.300 --> 00:45:15.900
I think they are going to play for a different number of years but the probability will be the same.
00:45:15.900 --> 00:45:22.800
Let us check that out but remember the numbers from this example.
00:45:22.800 --> 00:45:30.600
In example 5, again, each year the Long Beach jackrabbit has an 80% chance of winning the world pogo sticking championship.
00:45:30.600 --> 00:45:39.800
We want to find the expected number of championships that they will win in the next 5 years and the standard deviation in that total.
00:45:39.800 --> 00:45:41.400
Let us calculate that out.
00:45:41.400 --> 00:45:48.500
I’m going to use my generic formulas for expected value and variance, and standard deviation.
00:45:48.500 --> 00:45:52.900
You can find those formulas on the third slide in this lecture.
00:45:52.900 --> 00:45:55.200
If you do not remember those formulas, where they come from,
00:45:55.200 --> 00:46:01.600
just check back in the third slide of this lecture and you will see the following formulas.
00:46:01.600 --> 00:46:08.300
The expected value of Y is N × P, this is for the binomial distribution.
00:46:08.300 --> 00:46:14.500
The variance of Y is NPQ.
00:46:14.500 --> 00:46:21.200
The standard deviation which is always just the square root of the variance.
00:46:21.200 --> 00:46:26.500
That makes it in this case, the square root of NPQ.
00:46:26.500 --> 00:46:30.100
We are going to calculate each one of those for this particular problem.
00:46:30.100 --> 00:46:33.400
N, remember is the number of trials that we are running.
00:46:33.400 --> 00:46:39.100
In the previous example, we are running this over 7 years but now we are just running it over 5 years.
00:46:39.100 --> 00:46:45.600
I’m getting that from right here, that is my N, N = 5.
00:46:45.600 --> 00:46:50.900
The P here is the probability of winning any particular year and we are given that that is 80%.
00:46:50.900 --> 00:46:54.500
As a fraction, that is 4/5.
00:46:54.500 --> 00:46:58.700
I’m not going to go ahead and figure out what Q is.
00:46:58.700 --> 00:47:01.400
Remember, Q is always 1 – P.
00:47:01.400 --> 00:47:07.200
That is easy, that is 1 - 4/5 is 1/5.
00:47:07.200 --> 00:47:10.200
I think that is all I need to know for this one.
00:47:10.200 --> 00:47:12.700
Let us go ahead and calculate these out.
00:47:12.700 --> 00:47:23.900
The expected value, the expected number of championships that they will win over the next 5 years is N × P.
00:47:23.900 --> 00:47:33.400
N is 5, P is 4/5, that is just 4 championships.
00:47:33.400 --> 00:47:43.700
That of course should not be at all surprising because we are going to play for 5 years in a row.
00:47:43.700 --> 00:47:46.900
We have an 80% chance of winning in any given year.
00:47:46.900 --> 00:47:51.300
We expect to win 4/5 of the years.
00:47:51.300 --> 00:47:57.000
If we play for 5 years, we expect to win 4 out of 5 of those years on average.
00:47:57.000 --> 00:48:06.000
That is very intuitive result there but it is good that it is backed up by the formulas because probability can sometimes be counterintuitive.
00:48:06.000 --> 00:48:07.800
Let us go and find the variance.
00:48:07.800 --> 00:48:13.600
We are not being asked the variance in the problem but it is kind of a steppingstone to finding the standard deviation.
00:48:13.600 --> 00:48:15.400
It was worth finding the variance.
00:48:15.400 --> 00:48:25.100
NPQ is 5 × 4/5 × 1/5.
00:48:25.100 --> 00:48:29.900
The 5 and 1/5 cancel, that is just 4/5.
00:48:29.900 --> 00:48:33.300
That is the variance, that is not our full answer yet.
00:48:33.300 --> 00:48:36.100
To get our full answer, we want to find the standard deviation
00:48:36.100 --> 00:48:48.800
which is the square root of the quantity that we found above, √NPQ, which is √4/5.
00:48:48.800 --> 00:48:54.000
I could distribute that square root in the numerator and get 2/√5.
00:48:54.000 --> 00:48:59.300
Not a very enlightening number, I did go ahead and plug that into my calculator.
00:48:59.300 --> 00:49:05.100
What my calculator told me was that that is 0.894.
00:49:05.100 --> 00:49:17.700
That is the standard deviation in the number of championships we expect to win over a 5 year span.
00:49:17.700 --> 00:49:22.800
I will go ahead and box that up because that is our final answer there.
00:49:22.800 --> 00:49:25.300
Just to remind you where everything came from here.
00:49:25.300 --> 00:49:34.900
I got these formulas for expected value of variance and standard deviation straight off the third slide of this lecture series.
00:49:34.900 --> 00:49:37.000
You can just go back and look at those formulas.
00:49:37.000 --> 00:49:40.000
Those do correspond to the binomial distribution.
00:49:40.000 --> 00:49:44.800
Make sure you are working with the binomial distribution before you involve those formulas.
00:49:44.800 --> 00:49:50.100
This one is a binomial distribution because what is happening is that,
00:49:50.100 --> 00:49:54.500
the Long Beach jackrabbits are playing the championship year after year.
00:49:54.500 --> 00:49:56.800
Each year they either win or they lose.
00:49:56.800 --> 00:49:59.800
You can think of that as being almost like flipping a coin,
00:49:59.800 --> 00:50:04.600
except that it is not a 50-50 coin because the jackrabbits are dominant
00:50:04.600 --> 00:50:08.300
that every year the coin has an 80% chance of coming up heads.
00:50:08.300 --> 00:50:15.600
Every year, they have an 80% chance of winning and only 20% chance of losing.
00:50:15.600 --> 00:50:19.800
The expected value for binomial distribution we said was NP.
00:50:19.800 --> 00:50:27.400
Variance is NPQ, the standard deviation is always the square root of the variance, square root of NPQ.
00:50:27.400 --> 00:50:33.300
I’m just dropping in the numbers and get the numbers from the stem of the problem.
00:50:33.300 --> 00:50:36.300
N is the number of trials that you are going to run.
00:50:36.300 --> 00:50:42.500
In this case, that is the number of possible championships.
00:50:42.500 --> 00:50:48.800
We are going to play for 5 years, that N = 5 come from right there from the stem of the problem.
00:50:48.800 --> 00:50:57.000
P = 4/5 that comes from their chance of winning in any given year.
00:50:57.000 --> 00:51:00.500
80% translated into a fraction is 4/5.
00:51:00.500 --> 00:51:04.900
The Q is always 1 – P, that is 1 – 4/5 is 1/5.
00:51:04.900 --> 00:51:10.100
We just plot those numbers right into our formulas, the expected value is NP.
00:51:10.100 --> 00:51:15.000
It simplifies down to 4 championships which makes perfect intuitive sense.
00:51:15.000 --> 00:51:19.100
If you are going to play for 5 years and you got an 80% chance of winning each year,
00:51:19.100 --> 00:51:24.400
you expect to win about 4 championships on average.
00:51:24.400 --> 00:51:31.300
The variance NPQ simplifies down to 4/5 but that is not what we want.
00:51:31.300 --> 00:51:40.500
We want the standard deviation which is the square root of the variance, and that is 2 ÷ √5.
00:51:40.500 --> 00:51:47.000
I just threw that number into my calculator and it spat out the number 0.894
00:51:47.000 --> 00:51:55.900
is the standard deviation in the number of championships that we expect to win over any given 5 year span.
00:51:55.900 --> 00:51:59.600
That wraps up this lecture on the binomial distribution.
00:51:59.600 --> 00:52:03.500
This is part of the probability lecture series here on www.educator.com.
00:52:03.500 --> 00:52:05.600
Next up, we will have the geometric distribution.
00:52:05.600 --> 00:52:07.700
I hope you will stick around and learn about that.
00:52:07.700 --> 00:52:12.700
It looks a bit like the binomial distribution but there are certain key issues where it is different.
00:52:12.700 --> 00:52:19.400
In particular, you are not running a fixed number of trials anymore, you are running the trials over and over until you get a success.
00:52:19.400 --> 00:52:25.600
That turned out to change the probability distribution and it is going to change our mean and variance, and so on.
00:52:25.600 --> 00:52:29.500
We will look at that out in the next lecture, I hope you will stick around for that.
00:52:29.500 --> 00:52:34.100
In the meantime, as I said, these are the probability lectures here on www.educator.com.
00:52:34.100 --> 00:52:36.000
My name is Will Murray, thank you very much for watching, bye.