Bayes' Rule

Video Transcription

In this tutorial, you're going to learn about Bayes' rule. Bayes' rule is a theorem that allows us to turn around a conditional probability statement. It allows us to update or revise our probabilities in the light of new information. Essentially, what it does is it reverses the conditional probability formula. It allows us to find the probability of A given B from the probability of B given A.

So suppose that my favorite game show is Go for Broke. And a person rolls a die to determine a jar from which they select a chip. And the goal is to pick a green chip and win a fabulous prize. So if the chip is red, they leave with nothing.

And so, if a person rolls a one or a two, they select from jar A. And this is the one they'd like to go with because the jar contains nine green chips and 12 red chips. So it's a higher proportion than in jar B. That only contains five green chips and 25 red. But if they roll a three, four, five, or six, they have to select from jar B.

So this is the tree diagram. 1/3 probability of rolling a one or a two. 2/3 probability of rolling anything else. And nine out of 21 probability of getting the green chip once you're in jar A. And a 12 out of 21 probability of picking the red chip. Five out of 30 for the green in jar B. 25 out of 30 probability for the red in jar B.

By multiplying out the values on the tree diagram, we find that we get these values here-- 1/7 is the probability that you get the green chip from jar A. All of the other probabilities are listed here, as well. So suppose that it's just any random chip that I select. What's the probability that it came from jar A?

Well, I mean the question is, what's the probability that I have to pick from jar A? And it's 1/3. Not a particularly complicated problem, but suppose that I tuned in late to Go for Broke one day. And so I didn't get to see where the chip came from. And so I do see that the contestant won the prize by getting the green chip. But what's the probability that they actually picked jar A or rolled the one or two to get jar A, given that they got the green chip?

So what I'm actually trying to find is the probability that you would have picked from A, given that you got the green chip. So what you end up having to do is the conditional probability statement-- probability of A given G is probability of A and G over probability of G.

But the probability of A and G is one of the branches on the tree diagram. So it's probability of A times probability of G given A. And so what you're noticing is the probability of G given A helps us to find the probability of A given G.

When we actually go through and run the scenarios, this was one of the branches on the tree diagram. Probability of G-- G can happen in two ways. From jar A or from jar B. All of this fraction business simplifies down to the fraction 9/16.

So what that means is out of every 16 times you pick the green, nine of those times it came from jar A. And this is a surprising conclusion. Sometimes, Bayes' theorem can lead to surprising results, even though the probability that I picked from jar A was only 1/3. About over half of the winners, half of the green chips, about half the winners, over have the winners, come from jar A.

So the probability that you would pick from jar A, given that you ended up winning, is actually over half. So here's another example of when you might use Bayes' theorem. And this was actually used-- not on this particular plane route-- but this was used in an actual army practice to find some wreckage of a plane.

So suppose that a plane was going from Miami to Mexico City, but then it crashed somewhere in the Gulf of Mexico. They may have lost radio contact here. Now, not knowing what else might have happened with the plane, they might start searching in this area. Maybe a few days go by and people find debris over here.

Well, what Bayes' theorem allows us to do is analyze what's the probability that the plane crashed in this region, given that the debris was found way over here. If that probability is low, we might choose to revise our guess and start searching over here. Maybe they tried to go around the storm by going north.

The idea is that we can update our guesses based on new information. So we can update those probabilities. We thought the probability of it landing in this circle was high. Once we had new information, we decided that the probability that it landed in this black circle was low.

And so to recap. Bayes' rule is a mathematical theorem that allows us to amend probability statements based on new information. It's essentially our ability that we can turn around the conditional probability statement and find the probability of A given B from the probability of B given A.

So we can use newer knowledge to adjust probabilities of prior events. And essentially, what we're asking is, what's the probability that this first event happened given that the second event ended up happening.

So we talked about Bayes' rule. Good luck. And we'll see you next time.