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In this tutorial, you're going to learn about conditional probability. This means that what's the probability of something happening if something else is already happening? Let's take a look.
Conditional probability is the probability of some second event occurring, given that some first event has already occurred. It's written this way. Probability P bar A. The probability of B given A. This bar is pronounced, given.
So let's work an example. What is the probability of even on a roulette wheel? This is not a conditional probability yet because all I'm asking is what's the probability that it's even. All we need to do is count up all of the even sectors. Now zero and double zero don't count as even. They're counted as separate sectors.
So only the evens count. Where are the evens? The evens are in here, and in here. These are the ones that are even in black, these are the ones that are even in red. And there are 18 of the 38 numbers on the roulette wheel that are considered even.
This is a conditional probability statement. What is the probability that the sector is even, given that the sector is also black? Well, in this particular scenario, I'm telling you that it's black. So we're going to ignore all of the non-black sectors. We're going to ignore any of these that were neither black nor even, and the ones that were only even without being black. We're going to limit our scope on the Venn diagram to just the black ball.
Some of these in the black ball are also even. So if I'm telling you that you're in this ball, what's the probability of being even? Well, what we've done is we've limited the number of sectors that we can choose from. There's still several in here. There are 10 sectors in here that are even and black, whereas there are 18 total black sectors. So it's ten even out of 18 total black. Now these were both even and black. These were all the black.
So, let's examine that probability a little further. Ten out of 18. Where did the ten come from? The ten was the number that were both even and black. There were 10 sectors that did that. And 18 that were just black. So what we might be noticing is we might come upon a formula for the probability of E given B, or the general version with A's and B's in it.
The Probability of B given A is equal to their joint probability, A and B, like we did both even and black up here, divided by the probability of the thing you're given. I'm telling you to just look at the places where A is occurring. Now notice these ten and 18 were, in fact, not probabilities but raw numbers. But we can prove that it doesn't really matter which number we use. Big deal. We'll put some majesty around it.
All right, back to the issue at hand, where the ten and the 18 were actually frequencies. Ten was the number of sectors that were both even and black. And 18 was the number of sectors that were black, as opposed to probability. But if we look at the probability of being both even and black, the probability that a sector is both even and black, was 10 sectors out of 38 total sectors. And the probability of being a black sector was 18 out of 38.
What happened is the 38s end up cancelling off, and you end up with 10 over 18. So it doesn't really matter. You can put the probabilities in here, or you can put the frequencies in there directly.
And so to recap, conditional probability is the probability of some second event occurring, given that some first event has already occurred. It's calculated by dividing the joint probability of the two events by the probability of the existing event--the one that's already happening, the one that I'm giving you.
And this formula works for all events. This isn't a special formula that works only for independent events or only for mutually exclusive events. This is an event, or a formula rather, that works for all types of events. So this is a really nice one. And this is the conditional probability formula. Good luck and we'll see you next time.
The probability that one event occurs, given that another event has already occurred.