This tutorial talks about and probability for dependent events. As a quick review, this word dependent here, when we're talking about dependent events, that's when the occurrence or the outcome of one event affects the occurrence or outcome of the other event. So what happens in one place affects what happens in the other place. On the other hand, independent events are when the occurrences or outcomes of the event don't affect each other. There's no changes there, because one event has happened or because you know something that has happened.
When you have dependent events and you're trying to calculate the and probability, you need to use this general multiplication rule for probability. So this general multiplication rule is the overall rule for finding the probability of both A and B happening.
Now, this formula here is the general multiplication rule. So the probability of A times the probability of B, given that A. So this bar right here is set as given that. So you take the probability of event A happening, and you multiply that by the probability of event B happening, given that A has already happened. So this is where you're taking into account that dependent part of it.
Now, before we saw the special multiplication rule. That only worked on independent events. So this one will work on dependent events.
The nice thing is, though, that this general multiplication rule is still going to work, even if your events are independent. So it will work in both cases. It's going to work on dependent and independent events, and we'll see an example of each on the next slide.
So here I've kept the formula at the top. Kind of to remind us of what it is, our general multiplication rule.
Example 1 says, what is the probability of drawing a red card followed by drawing a spade? Now, in this example, event A is drawing a red card and event B is drawing a spade. So first, we want to figure out if these are dependent or independent events. And they are, in fact, dependent.
Once you've taken out that red card from the deck, the probability of getting a spade changes. So the outcome, or the event here, is affecting this one, so we have dependent events. So we would want to use this multiplication rule.
So now first, what's the probability of drawing a red card, that probability of event A? Well, there's two colors. There's red and there's black. And drawing a red card, it's one of those two colors. So we have 1/2.
You could have also thought about the fact that there are 52 cards, and we're pulling out 26 of them could be red, so 26 out of 52 is going to simplify to 1/2.
Now, we're going to multiply by the probability of B happening, given that A has already happened. So what's the probability of drawing a spade given that we've already drawn a red card?
So now, once we've taken out that red card, instead of 52, we only have 51. And how many spades are there? There are 13. And none of those spades could have been drawn out before, because the spades are black, and we took a red.
So now when we multiply these together, we have our and probability for the dependent events. So 1 times 13 is 13. 2 times is 51 is 102. So our probability of drawing a red card followed by drawing a spade is 13 out of 102.
Now, in the second example, it says, what is the probability of rolling a 5 and flipping heads? Now, these two events here are independent. When you're rolling a six-sided die, it doesn't really matter and it doesn't affect what happens when you flip a coin. So we could use the special multiplication rule here. It's going to give us the same thing.
So the probability of rolling a 5, well, we have a six-sided die and 5 is one of those six options, so 1 out of 6. And then flipping heads, so the probability of flipping a heads given that you've rolled a 6, well, that's just the same old normal flipping heads probability, so 1 out of 2, because the rolling of a 5 doesn't affect what you flip.
So here, 1 times 1. 6 times 2 is 12, and we have a 1 over 12 probability. So we'd get the same thing if we just used the special multiplication rule from before, because we would again be multiplying 1/6 times 1/2, because these events are independent. So the rule works for whether it's dependent or independent.
This has been your tutorial on and probability for dependent events.