 Alright, so another rule concerning basic probabilities that we need to understand is the rule that concerns independent events. So definition of an independent event from a probability point of view is when the outcome of one event doesn't have any effect positive or negative or any sort on another event. So the standard example of that would be flipping coins. If I flip a coin, I either get heads or tails. If I then flip another coin, a different coin, or the same coin a second time, whatever I got on the first coin flip doesn't have any effect on what I get on the second flip. I'm not anymore or less likely to get heads on the second coin flip depending on whether I got a heads on the first coin flip. So for example, if I ask in that case if I flip two different coins and I want to know what's the probability that I get heads twice in a row, what's the probability that I get heads on the first coin flip and I get a heads on the second coin flip? The probability that I get heads on the first flip is one out of two. I have two options, heads or tails, and I've got a 50-50 chance of getting heads. I have to get heads on the first flip and I have to get heads on the second flip. So as May had seemed intuitive from the moment I asked the question, the odds of getting heads on both coin flips, the first one and the second one, is one out of four because I have 50-50 chance on the first one and a 50-50 chance on the second one. So only one of the four, I could have gotten heads heads, heads tails, tails heads, tails tails, only one of those four outcomes gave me heads on both of those coin flips. So that's an example of the more general rule for independent events, which is if I want to know what's the probability of A happening and some other event B happening, then I just take the two probabilities and multiply them together. As long as A and B are both independent events, they don't affect one another. So to work another example that involves independent events, let's say we do something with cards this time. I have an ordinary deck of 52 playing cards and I draw one card from that deck. Let's say I want to know what's the probability that I draw a queen of hearts out of the deck. So one way of thinking about that, of course, is there's only one queen of hearts and there's 52 cards. There's a one chance in 52 that I would draw a queen of hearts out of the deck. Or I could think of it as asking what's the probability that the card I draw is a queen and the card I drew is also a heart. So from that point of view, we can think about as two independent events. Whether I draw a queen or not doesn't have any bearing on whether it's a heart or not. I could also have thought about that problem as what's the probability that I drew a queen multiplied by the probability that I drew a heart. And there's four, well maybe the easier way to think about it is there's 13 different denominations of cards. Only one of them is a queen. So the chances are one out of 13 that I'm going to draw a queen. There's four different suits. Only one of them is hearts. So there's a one chance in four that I'm going to draw a heart. So if I multiply those two fractions together, one-thirteenth times one-fourth, then what I get is one-fifty-second. Again, there's more than one way to do the problem. Either way of thinking about it gives you the right answer, but it's certainly possible to think about this problem as a probability of two independent events because the two events are independent. So again, the basic rule here is whenever you have an AND problem, if you want one thing to happen and another thing to happen, what you do is you multiply the two probabilities together. That's distinct from the mutually exclusive events or problems where you add the two probabilities. But there's one very important caveat in this case. You can only multiply the probabilities for an AND event if the two events are independent. And this is more easily easy to trip over than the caveats for the other basic probability events because many times events are not independent. So it's always worth stopping and thinking about whether two events are in fact independent of one another. And to give you an example of how that can trip you up, I'll work one more example, which is to say, let's pick a random day of the year. So you can, for example, think of the day of your last birthday. So that's a calendar day on the year, and I don't know what that day that is. So from my point of view, I've just picked a random day by asking you what your birthday is. So since I don't know what your birthday is, if I ask, what's the probability that the date that I just randomly picked your last birthday was somewhere in June, July, August, or September. So it turns out that if we count up the days, that 122 of the days in the year are in June, July, August, and September. So there's a 122 out of 365 chance that the date we've now picked together has fallen in the range of June through September. I could also ask on that date of your last birthday, what's the probability that it was cold? If I define cold as if it got below freezing on that day, turns out at least at Clemson University, where this is being filmed, there's 62 days on average every year that drop below freezing. So the probability of that date that we're talking about being a cold day being below freezing, there's a 62 out of 365 chance that that randomly selected date was cold. If I now ask what's the probability that the date we've picked is in this range June through September and that it was cold, then I'll put a question mark here because of course this is going to turn out not to be true. I can say is that perhaps the probability of it being June through September multiplied by the probability that it was cold, in other words, is it 122 out of 365 multiplied by 62 out of 365? So that would work out to be if this is the right answer. So 122 out of 365, roughly one out of three days is in this summer time period, roughly one out of six is cold. If I multiply those two together, what I actually get is a number close to 5.7%. But of course you know that's not true. There's hardly any days in this range of June through September that are in fact below freezing. And in fact, if I look at the weather records, the same place that I got this record of cold days, there has never been a day in June through September in Clemson, South Carolina that was below freezing. So this is not the right answer. So this question mark doesn't hold. We can't multiply the two probabilities together. This is not the right answer. The correct answer to this question is zero. There's no chance that the date was both below freezing and in June through September, which we can only calculate with some extra information by going and looking at the weather records, for example. But the reason that this multiplying the two individual probabilities doesn't work in a case like this is not because this rule is incorrect. It's because this rule doesn't apply because the rule only applies if the two events are independent and whether a day is in the summer and whether a day is below freezing, those are not independent events. There's a correlation. So if there's a correlation between A and B, if A is correlated with B, or if knowing whether it's a date in the summer has some effect on whether the day is cold or not, then those events are correlated and we can't use this independent event formulation. So this one is just a reminder that while it's going to become very tempting anytime you see an and question, probably a something and something else, it's going to be very tempting to multiply two probabilities together. It's always worthwhile to stop and ask yourself, are these events independent? If so, then go ahead and multiply probabilities and the question will proceed fairly simply. But if the two events are correlated, then we will have to think about it a little bit harder.