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  • Hi, I'm Adriene Hill, and welcome back to Crash Course Statistics.

  • When you're using the patterns of probability to predict events in your life, it can be

  • nice to have some mathematical shortcuts.

  • That way, when you're reading People Magazine, you won't have to spend so much time sidetracked,

  • calculating how likely it is that Harry and Meghan will have 3 boys and 2 girls if they

  • have 5 kids.

  • And you won't have to pause Willy Wonka and the Chocolate Factory, right as Grandpa

  • Joe starts singing, to figure out how likely it is that you could have found one Golden

  • Ticket by opening 4 chocolate bars.

  • Today, we'll introduce you to some of these shortcuts so you can get back to the movie.

  • Before we start today, heads up -- these shortcuts are equations.

  • And we try in this series not to spend too much time plugging numbers into equations,

  • but today we felt like we kinda needed to.

  • Because understanding these things is important.

  • So to sweeten the deal, we added some zombies.

  • INTRO

  • You're in your kitchen, making a piece of toast with your old toaster--which has seen

  • better days.

  • You've figured out that each time you make a piece of toast you have a 20% chance of

  • being shocked--not lethally, but painfully.

  • Alright so maybe it's time to get another toaster.

  • But you haven't had a chance.

  • Anyway you eat toast every weekday morning (you eat pancakes on the weekends), and you're

  • wondering how many shocks you'll get this week.

  • It's a stressful week and you've decided that toast is only worth the risk

  • if you'll probably get shocked only once.

  • You know the multiplication rule of probability, so you can calculate this.

  • There are five different ways you can receive only one shock this week: you either get shocked

  • once on Monday, Tuesday, Wednesday, Thursday, or Friday and then remain shock free on the

  • 4 other days of the week.

  • If we represent a shock with an X and a non-shock day with an O, the possibilities for your

  • week look something like this.

  • Now we need to calculate the probability of getting one shock and four non-shocks using

  • the multiplication rule.

  • Let's look at the probability of only getting shocked on Monday.

  • The probability of getting shocked is 20%, so the probability of not getting shocked

  • on Tuesday is 80% and similarly on Wednesday through Friday there's an 80% chance each

  • day of not getting shocked.

  • So the probability of getting shocked on Monday and not on Tuesday-Friday is

  • 0.2 x 0.8 x 0.8 x 0.8 x 0.8 = ~0.082.

  • That's about an 8.2% chance that you'll get shocked on Monday and not for the rest

  • of the week, so now we've got to calculate the rest of the one-shock options.

  • The probability of getting shocked only on Tuesday is the same, about 8.2%, since order

  • doesn't matter in multiplication.

  • The probability of the Monday-only option or the Tuesday-only option or any of the remaining

  • 3 options can be calculated using the addition rule.

  • You have 8.2% + 8.2%+ 8.2%+ 8.2%+ 8.2% also known as 5 x 8.2% chance of just one shock.

  • That's a 41% chance of getting shocked only once this work week, so you decide to risk it.

  • That was a lot of work just to figure out whether it's a good idea to have toast,

  • and thankfully there's a more compact formula.

  • The formula, called the Binomial Distribution formula, takes the math we just did and simplifies it.

  • In our toast example, we first figured out the probability of only getting shocked once.

  • To do this we multiplied each day's probability together.

  • Let's use exponents to make this formula a little bit shorter.

  • We can combine all the 0.2s and all the 0.8s to give us this.

  • That's about 8 percent..

  • Notice that the two exponents add up to 5--the total number of days.

  • This formula works for finding out the chance of getting shocked only once, but we can also

  • use it to find out the chance of getting shocked another number of times.

  • In general the formula looks like this:

  • For example, the probability of getting shocked only on Tuesday and Wednesday would be this.

  • = ~0.02 or 2%

  • We also need to account for the number of ways that getting shocked once or twice in

  • a week can happen.

  • To do this, we'll use a very useful formula--the Binomial Coefficient Formula--from a field

  • of math called Combinatorics.

  • The Binomial Coefficient formula makes it easy for us to find out how many ways a certain

  • ratio of successes--not getting shocked--to failures--getting shocked, can occur.

  • In a general form, it looks like this: (n-Choose-k) but you also can read it aswe have n things,

  • count the number of different ways we can choose k of them”.

  • For our toast example (5-Choose-1 is 5).

  • That is there are 5 different ways we can only receive one shock from our toaster this week.

  • The math behind this snazzy formula uses a lot of factorials which are the product of

  • an integer and all the integers less than it, and looks like this:

  • You remember factorials because they're the ones that look like they're always shouting.

  • See those exclamation marks?!

  • We won't dig into this formula here, but for more information, we'll link some resources

  • in the description.

  • Now we have all the pieces of our binomial distribution formula, let's put them together.

  • First the Binomial Coefficient which tells us how many ways we can have one shock and

  • four non-shocks, and then our shorthand multiplication of probabilities.

  • We put it together and we have a full blown formula for calculating the probability of

  • getting shocked on one out of five days this week, about 40%.

  • It took us a while to get here, but we now have a general formula for calculating any

  • similar probability.

  • If p is the probability of our event happening then here's our formula:

  • For example, supposing that there is an equal chance of having a boy or girl, if we want

  • to find out the probability of a couple having 3 girls out of 5 children, we can simply plug

  • our numbers into the equation.

  • We do that and we see there's about a 31% chance of having 3 of their 5 children be

  • girls.

  • And now we get to the zombies: it's the beginning of the zombie apocalypse and you, thankfully, still

  • have your brains.

  • But there's a bunch of people between you and the nearest shelter, and you want to know

  • how likely it is that none of them have been bitten and infected with the zombie virus.

  • Since you can't always tell if a person is infected right away, you decide to use

  • your binomial probabilities to calculate your chances.

  • It's early stages, so right now there's only about a 5% infection rate in the population.

  • That means the probability of someone being a zombie is about 5%, and the probability

  • of not being infected is still about 95%.

  • Peeking out at the crowd that stands between you and the shelter you count 20 people.

  • Plugging these numbers into our formula, we can see that there's about a 36% chance

  • none of the people you'd encounter on your run to safety will try to eat your brains.

  • Those are pretty good odds, and you're fast.

  • You know that you could probably safely reach the shelter even if there were one or two

  • zombies, so let's calculate those probabilities as well.

  • There's about a 36% chance of meeting no zombies, about a 37% chance of having only

  • one--easily dodgeable-- zombie, and about an 18% chance of having to out maneuver only

  • two, that means that cumulatively, that's about a 91% chance that you'll be able to

  • sprint your way to safety.

  • We could calculate the probability for every possible number of zombies from 0 to 20.

  • If we did that, we would get the discrete distribution for this specific type of problem

  • where there are 20 events, and the probability of the event of interest--here, the zombie

  • infection rate--is 5 %.

  • But in a more general sense, I also want to know how many zombies I would expect to confront

  • on average - just so I could prepare.

  • I could try to guess the mean by looking at the graph of the binomial distribution for

  • all of the possible cases.

  • Just by eyeballing it, I'd say the mean of this distribution is around 1, and it is!

  • The actual formula to find the mean of a binomial distribution is n--the number of events we're

  • looking at--times p--the probability of what we're interested in, the probability of

  • being a zombie.

  • Since the probability of being a zombie is 5%, it makes sense that on average, about

  • 5% of any population will be zombies.

  • Since we have a group of 20, we expect about 5% of 20--or 1--zombie infection on average.

  • While zombies are probably not on your day to day list of concerns, this kind of calculation

  • might be.

  • Important public health issues such as the spread of pandemic-level viruses can been

  • modeled using a similar approach.

  • We could just as easily calculate the probability that of the 40 people you shook hands with

  • at that Zombie Apocalypse meet-up, 2 or fewer had the cold that has been going around.

  • Or we could calculate the mean number of cold infected people you could expect to shake

  • hands with at any given meeting.

  • If the probability of having a cold at that given moment in the population is 20%, the

  • probability of only 2 or fewer people having a cold is only about 0.7%.

  • From the binomial distribution with 40 people and a 20% chance of having a cold we can see

  • that it's MUCH more likely that more than 2 people will have the sniffles, in fact you'd

  • expect 20% of 40 --or 8 people--to have a cold at any similar meeting with 40 people.

  • A special case of the Binomial Distribution with only one trial--or just one event, a

  • single coin flip, for example--is called the Bernoulli Distribution.

  • It represents the probability of of getting either a success or a failure.

  • Our outcome--x--can either be a 0 (failure) or a 1 (success).

  • The general formula for the Bernoulli distribution where p is the probability of success is this.

  • And by plugging in 1 or 0, we can see that the probability that our outcome is a success

  • is p, and the probability of a failure is (1-p).

  • For example, if the probability of rolling an odd prime number ( 3 or 5) on a single

  • dice roll is about 33.33%, then the bernoulli distribution for rolling a prime number would

  • be the probability of rolling an odd prime to the x power multiplied by the probability

  • of not rolling an odd prime to the one minus x power.

  • When x is 0--a failure-- then the probability is 1 minus one third or around 66.66%, and

  • when x is 1--a success--the probability is around 33.33%.

  • Though relatively simple, the bernoulli formula is a useful building block.

  • For instance, you can think of the binomial probabilities that we did as many bernoulli

  • trials one after another.

  • And in reality, that's what they are!

  • Alright let's get back to the zombies.

  • Things have gotten much much worse.

  • Zombies are everywhere - you're injured really badly.

  • You need a blood infusion and need so much blood that you'll need some from all 3 of

  • the people around you, luckily you're type AB positive so you can get blood from any

  • blood type, but you have something else to worry about: the latent zombie virus.

  • All three of your buddies seem okay, but you heard on the radio the population now has

  • a 30% infection rate.

  • So there's a 30% chance they're infected with the zombie virus and are just asymptomatic.

  • Luckily, the human immune system can handle the virus in very small doses, so you'll

  • be okay if at least two thirds of the blood is clean.

  • What are the chances you're gonna survive the infusion?

  • If someone has a latent zombie infection,then the bernoulli trial would have one success

  • and zero failures, giving this formula for the trial:

  • Which as we expect, means that there's a 30% chance that this one random person has

  • a latent zombie infection.

  • And every person who is not infected is considered to be one failure and 0 successes, which has

  • a 70% chance, based on this bernoulli formula:

  • We can use different combinations of these two bernoulli formulas and multiplication

  • to figure out the probability that you'll remain uninfected after your much needed blood

  • transfusion.

  • You'll be fine if no one has any latent zombie virus, and we can calculate the probability

  • of that happening by combining three successful bernoulli trials.

  • And this looks a lot like the second part of our binomial formula, because it is!

  • A 34.3% chance of survival is kinda low, but don't forget you'll also be okay if only

  • one person has a latent infection, in other words two successes and one failure, which

  • we can combine like this.

  • And there's three ways this could happen (any of your three friends could be infected)

  • so theres a 3 times 14.7--or a 44.1% chance that only one of your friends will be infected.

  • Combining the 44.1% chance that only one friend is infected, with the probability that all

  • their blood is zombie virus free, we can see that there's a 78.4% chance that this life-saving

  • transfusion will go your way.

  • You're not going to be a zombieprobably.

  • These two formulas are shortcuts to help us spend less time calculating, and more time

  • applying probabilities to our lives.

  • These concepts also allow us to see how probabilities of a single event can give surprising results

  • when we combine a bunch of them.

  • For example let's say the probability of NOT having a flat tire each year is 95%, meaning

  • there's a 5% chance that you'll get a flat tire.

  • But over 15 years, that's only a 46% chance your tires are going to stay inflated, that

  • means there's a 54% chance of getting a flat tire at some point.

  • We don't see the world as just one event at a time - many things are connected; So

  • having the tools to think of many probabilities together is important.

  • Thanks for watching.

  • Stay zombie virus free, and I'll see you next time.

Hi, I'm Adriene Hill, and welcome back to Crash Course Statistics.

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二項分布クラッシュコース統計学 #15 (The Binomial Distribution: Crash Course Statistics #15)

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    林宜悉 に公開 2021 年 01 月 14 日
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