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We've talked about the probability distributions for both discrete random variables and continuous random

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variables.

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Now we want to drill down a little further into a specific type of discrete random variable, which

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is the binomial random variable.

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So a binomial random variable for now, we'll just call it the discrete random variable x, but it's

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a special type of discrete random variable that meets these four criteria.

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We often indicate the binomial random variable like this with these parameters and and p, so we might

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write X of NP or B of an P for binomial.

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We'll see it this way as well.

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But the binomial random variable is always modeling what we call the number of successes in some sequence

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of NW independent experiments.

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That's where that first criteria comes in independent trials.

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So let's take the simplest example of flipping a coin.

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When we flip a coin multiple times and we call each coin flip one trial, those are independent trials

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because each coin flip is not affected by any of the other coin flips, whether we flip heads or tails

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on one flip is totally independent of whether we flip heads or tails on another flip.

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So we have independent trials.

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We define the outcome of each trial as a success or a failure.

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And these ideas here of success or failure don't really have anything to do with indicating some measurement

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of good or bad.

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By success, we just mean the outcome that we're investigating or the outcome we're looking for.

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So if we're looking at the number of times that we flip heads, we would just define flipping heads

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as a success and anything else as a failure.

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In this case, the only other possible outcome is flipping tails.

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So flipping tails would be a failure.

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Flipping heads would be a success.

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But if we're investigating the number of tails that we flip instead, then we would simply define flipping

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tails as the success and everything else, in this case flipping heads as the failure.

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So we define the outcome we want as success and everything else as a failure.

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We have to be able to categorize outcomes in that way, which is partly why this can only be a discrete

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random variable.

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In order to model this situation with a binomial random variable, we also have to have a fixed number

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of trials, so we can't use a binomial random variable to model a situation where we want to look at

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probability.

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When we flip a coin an infinite number of times we have to specify we're flipping the coin ten times

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or we're flipping the coin 50 times.

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The number of trials has to be fixed.

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We have to predetermine how many times we're going to run this trial.

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And that fixed number of trials is this number.

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And here.

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So this is a fixed number of trials.

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RN And then the probability of that success that we defined earlier has to be constant.

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So with the coin flip for every trial, for every time we flip the coin, the probability of success

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is always one half or one over two or 50% or 0.5, and that stays constant because each coin flip is

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independent.

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The probability of success can't change from trial to trial.

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If we want to model a situation with a binomial random variable, it has to stay constant and that probability

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of success is this value.

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P So when we see a binomial random variable expressed as B of NP or X of NP, these values indicate

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the number of trials that were performing and then the probability of success on any particular trial.

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Remember too, that total probability always sums to one or 100%, which means that if we define the

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probability of success as P, then the probability of failure has to be one minus P, all of the other

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probability other than this probability of success.

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And so we often define failure as one minus P, and sometimes we write that as Q So Q is one minus P,

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This is the probability of success.

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P And the probability of failure one minus P, sometimes written as.

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Q So these are the criteria that we have to meet in order to model our situation with a binomial random

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variable.

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Keep in mind that we don't just have to limit ourselves either to a situation where there are only two

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outcomes, like with a coin flip, there's only two outcomes heads or tails.

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We can have more than two outcomes.

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So for instance, if we are rolling a six sided die and we want to define success as rolling a two,

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we would just define success as rolling a two.

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And then failure is rolling a one, three, four, five or six.

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It's rolling anything other than a two.

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Our probability of success would of course be one over six because the probability of rolling a two

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is one over six, and so the probability of success would be one over six.

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The probability of failure would be five over six.

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So not only can we have more than two possible outcomes, like with a die roll, but that probability

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of success doesn't have to be exactly one half.

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With the coin flip, we have just two equally likely outcomes heads or tails.

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And.

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The probability that either of those show up is equivalent.

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It's one half for heads and one half for tails, but we can have more than two outcomes, and the probabilities

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of success and failure don't have to be equivalent.

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So we could use a binomial random variable for a coin flip because each coin flip is independent of

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all other coin flips.

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We can define the outcome of each trial as a success or a failure.

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If we're looking at how many times we flip heads, then we would define heads as a success and tails

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as a failure.

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We can define a fixed number of trials.

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Let's say we're flipping a coin at 20 times and the probability of success remains constant because

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the probability of flipping heads remains one half for every coin flip.

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But we could also use this to model a die roll If, let's say we define success as rolling a two because

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every die roll is independent, the role of the die won't be affected by any of the previous roles.

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We can define our success specifically.

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So let's say we want to roll a two.

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Then we would define rolling a two as a success and rolling anything else as a failure.

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So rolling a1a3 of four of five or a six would be a failure.

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We can define a fixed number of trials.

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So let's say we roll the dice ten times and the probability of success remains constant.

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The probability of rolling a two always remains one over six.

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That probability remains constant for every trial, for every die roll.

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Contrast this with something that doesn't meet the criteria for a binomial random variable, like pulling

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cards from a deck of cards without replacing the cards.

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So if we're pulling cards from a standard 52 card deck and we're not replacing them, then right away

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we know that our trials are independent because when we pull the first card, we start out with 52 cards

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in the deck.

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And so the probability of pulling any one card is going to be one over 52.

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If we don't replace that card and we go to pull a second card, then the probability of pulling any

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of the remaining cards from the remaining deck is going to be one over the 51 remaining cards.

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So right away, the trials are not independent.

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The probability of success is going to be changing.

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We could not model that kind of scenario with a binomial random variable Now that we know when we can

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use a binomial random variable, let's talk about the probability associated with this kind of variable.

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What we're interested in here is given by this formula.

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This is the probability of K successes in N attempts.

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So remember we said that we had a fixed number of trials.

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N What we want to be able to calculate is the probability that we come up with K number of successes

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in N number of fixed trials.

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So if we're rolling a die 20 times and we want to know the probability that we roll heads 13 times,

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then we would calculate here the probability of K equals 13 in N equals 20.

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And the way that we calculate that is with this binomial coefficient here.

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We talked about this earlier when we looked at permutations and combinations.

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So the formula, again as a reminder for this binomial coefficient is K successes in n attempts.

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It's the combination and choose K and we calculate it as nn factorial divided by k, factorial times

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the quantity and minus k factorial.

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This is just a combination, so we have to calculate this value first.

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Plug it into this formula here for the binomial coefficient and then we multiply that by p the probability

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of success for our binomial random variable raised to the power.

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This is the exponent here of K, the number of successes that we're looking for, and then we multiply

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that by this one minus p which remember we looked at here is the probability of failure one minus P,

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we also call it Q It's the probability of failure.

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Where P is the probability of success.

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So one minus P is that failure probability and we raise that to the power of N minus K, where n again

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is the number of trials and K is the number of successes we're looking for.

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So let's just say to take an example that we're running a rideshare company, we're looking at a particular

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city where we operate our ridesharing company and we want to define success as one of our drivers arriving

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within 10 minutes of the moment when the passenger has requested the ride.

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We're interested in that figure because we don't want our passengers waiting longer than 10 minutes

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from the moment when they request a ride through our app.

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So we're going to define each trial as one ride.

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We're going to define success as a driver arriving within 10 minutes and failure as it takes longer

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than 10 minutes for the driver to arrive.

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We're just going to look at five trials or five rides.

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So we're going to randomly select five rides from our database and we're going to say that from the

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data we already have about the rides we've been operating in this city, that the probability of success

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for any particular.

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Ride is 75%.

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So in the past, 75% of passengers who request a ride have their driver arrive within 10 minutes of

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the request.

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So PE is going to be 0.75.

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So in that circumstance we meet these conditions for a binomial random variable to model this situation

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with a binomial random variable.

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And now let's say we want to look at the probability of finding K successes in N attempts.

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So we set n was going to be five.

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We're going to choose five rides randomly from our database.

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What's the probability that zero of those five have the driver arrive in 10 minutes or less?

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What's the probability that one in five arrives in 10 minutes or less?

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So in other words, this probability that we're going to be calculating allows us to understand, basically,

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if we choose five rides at random, what's the probability that the driver will arrive within 10 minutes

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for zero of those rides?

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One of those rides, two of those rides all the way up to five of those rides.

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So here's what some of those calculations would look like.

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The probability that zero drivers arrive within 10 minutes if we pull a random sample of five rides

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would be calculated this way.

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We have this binomial coefficient and choose k or five choose zero.

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We're selecting five rides and we want the probability that we have zero successes.

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Zero drivers arrive within 10 minutes and then we take our probability of success.

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P which we said was 0.75.

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There's a 75% chance that a driver arrives within 10 minutes.

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So we raise 0.75 to the power of zero because we're looking here for zero successes, K successes.

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And then we take our probability of failure, which is just one minus P, So one -0.75 is 0.25 and we

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raise that to the end minus K.

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In this case, that's five -0 or five.

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And we can use a calculator or software to calculate an approximate decimal value.

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Here we've rounded all of our decimal values to six decimal places.

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Then this second line here calculates the probability that of five rides.

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Exactly.

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One driver arrives within 10 minutes and we could continue calculating the probability that two of the

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five arrive within 10 minutes.

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Three of the five, four of the five all the way down to the last option, which is that of five rides,

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all five drivers arrive within 10 minutes of the ride request.

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So we could build out that table of probability.

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We could get a value for each of those number of successes.

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k012, three, four, five And then we can use this information to build out the probability distribution

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for a binomial random variable.

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Remember, since that binomial random variable is a discrete random variable, it's a special kind of

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discrete random variable.

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Our probability distribution will be a probability mass function that looks like this.

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Here is the probability distribution for this specific scenario with the ride sharing company.

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This should remind us of the general form of a probability distribution, a probability mass function

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for a discrete random variable along the horizontal axis.

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Here on the bottom, we place all of the discrete countable possible outcomes for our variable.

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These are the numbers of our possible successes.

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We can have zero successes, one success, two successes, all the way up to five out of five successes.

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We see these values here coming from these values of k01, two, three, four, all the way up to five.

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So we put those values along the horizontal axis and then along the vertical axis for our probability

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mass function, for our special type of discrete random variable.

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We plot the probability of each of these particular values along the horizontal axis.

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So the probability that we got zero successes in five attempts was almost zero.

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We found 0.000977.

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That probability is so small that our little bar of probability doesn't even appear here.

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The probability that we get one success in five attempts is just over 1%.

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And so we see here if this is 10% and this is 5% along our graph, we see here our 1% value, about

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a 1% chance that exactly one of our five drivers arrives in 10 minutes or less and we could continue

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plotting all of these probability values all the way up to the probability that all five of the five

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drivers arrive in 10 minutes or less.

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That probability for all five drivers arriving for five successes is just under 24%.

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And we see that value here.

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So we have here 20%.

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And then this line here representing 25% or 0.25, and we see our just less than 24% probability right

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under that line there.

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So this.

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Then becomes the probability distribution for this specific binomial random variable.

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The probability distribution will look different for every binomial random variable depending on the

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values of N and P.

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So in this scenario we chose N equals five, but if we chose instead, N equals 20 to run 20 trials

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to look at 20 different rides that take place in our app, then instead of just the values zero through

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five along the horizontal axis, we would have the values zero through 20.

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And then the exact distribution here depends on the fact that our historical data told us that we had

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a 75% chance of success, but maybe in a different city.

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Our chance of success or chance that a driver arrives within 10 minutes is 60% or 90% or only 20%.

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Whatever value we have for that chance of success, of course, that is going to affect the shape of

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our distribution.

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So this distribution is unique to n equals five trials with a P equals 0.75 chance of success.

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But that is how we build the probability distribution.

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And of course, once we build it, it should roughly make sense to us.

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If we run five trials or five experiments and our expected probability of success is 75%, well, what's

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75% of five?

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It's 3.75.

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And so roughly, we would expect that most of our mass here, in our probability, mass function in

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this distribution would be clustered around the 3.75 mark.

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And we see that that's roughly true.

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Most of the weight of our distribution is clustered around these three, four or five values.

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More specifically, even the four value, which of course is the value closest to 3.75.

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So very roughly with some general intuition, this distribution looks like it could model our scenario

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where we have five trials and a 75% chance of success.

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That being said, let's get a little bit more specific and look at the properties of A binomial random

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variable B of P, And let's say here that yes, in fact the calculation we just did is the calculation

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that we use for the mean of a binomial random variable.

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So once we know that we have a binomial random variable, it meets these conditions, we're going to

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use a binomial random variable to model our scenario.

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Then we unlock all of these formulas that apply to the binomial random variable or the binomial distribution.

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Here the mean is always going to be end times p.

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In our case, five times 0.75 or 3.75.

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So our mean our expected value is 3.75.

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So if we randomly select five rides, then we expect roughly 3.75 of them.

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In terms of sort of this long term average, we expect 3.75 of those five rides to have the driver arrive

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within 10 minutes.

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Of course, this is a discrete random variable.

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So out of five rides we'll never actually see exactly 3.75 rides meet.

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That criteria will always see either three or four or five or two or one or zero.

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We'll never see exactly 3.75, but that gives us a long term average.

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It gives us our expected value or our mean.

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So we can always find that as n times p in this case five times 0.75.

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And then the variance here is given by NP times one minus P.

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Remember sometimes two, we express this one minus P probability of failure value as Q.

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So we also see this written sometimes as NP.

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Q In our case, that's five times 0.7, five times 0.25 or 0.9375.

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That's going to be the variance of this binomial distribution for the binomial random variable.

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And then standard deviation is always just the square root of variance.

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So for standard deviation, we would take the square root of 0.9375, and that's approximately equal

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to if we round 0.9682.

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And so now we have our mean variance in standard deviation for the binomial distribution.

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So to summarize, let's just say that we always have to start with these four conditions.

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We have to meet these four conditions in order for a binomial random variable to even be on the table.

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We can't model a situation as a binomial random variable unless we're meeting these four criteria.

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So we always have to go through and verify that each of these four criteria is holding.

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If we're meeting all of these criteria, then we can use a binomial random variable which then unlocks

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for us all of these probability calculations.

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We can specifically find the probability of K successes in N trials using this formula here.

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Here where this is the binomial coefficient that we calculate this way.

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We learned about this when we talked about permutations and combinations, so we can calculate the probability

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for every possible value of K, and K is always going to take on values between zero and N.

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So if n is our number of trials, K is always going to be given by the set of integers between zero

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and N, so k can be equal to zero and then it can be one, two, three, four, five, six, etc. all

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the way up to whatever our value of DN is.

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So we can calculate probability for all of those different values of K, and then we can use those probability

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values to build the binomial distribution.

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This visual representation of the probability associated with our binomial random variable.

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And then regardless of whether or not we've already calculated all of this probability or built this

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binomial distribution, even if we don't have either of those things, all we need to find the mean

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variance in standard deviation of this distribution are the values of GN and P, which we have from

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our criteria that we unpacked at the beginning.

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We know our fixed number of trials.

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NW and we know the probability of success.

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P And those are the only two values we need to find mean variance in standard deviation so we can go

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directly to calculating these values even before we've calculated all of this probability and used it

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to build this binomial distribution.