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 as “we 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 zombie… probably.
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.