You'll become familiar with the concept of independent events, or that one event in no way affects what happens in the second event. Keep in mind, too, that the sum of the probabilities of all the possible events should equal 1.Created bySal Khan.
Let's think about the situation when we have a completely fair coin here let me draw, I'll assume it's a quarter or something, so this is a quarter, let me draw a faint attempt at a profile of George Washington, well that will do. It's a fair coin and we're gonna flip it a bunch of times and figure out the different probabilities. So let's start with a straight forward one. Let's just flip it once, so with one flip of the coin, what's the probability of getting heads? Well, there's 2 equally likely possibilities and the one with heads is 1 of those 2 equally likely possibilities, so there's a one half chance. Same thing if we would of asked what is the probability of getting tails? There are two equally likely possibilities and one of those gives us tails, so one half. And so this is one thing to realize: if you take the probabilities of heads plus the probabilities of tails, you get one half plus one half, which is one, and this is generally too: the sum of the probabilities of all of the possible events should be equal to 1, and that makes sense, 'cause you're taking, you're adding up all of these fractions , the numerator and then add up to all of the possible events the denominator is always all the possible events, so you'll have all the possible events over all the possible events when you add all of these things up. Now let's take it up a notch. Let's figure out the probability of ... I'm going to take this coin and I'm going to flip it twice. ... the probability of getting a heads and then getting another heads. The probability of getting a head and then another head. There's two ways to think about it . One way is just to think about all of the different possibilities: I could get a head on the first flip and a head on the second flip, head on the first flip, tail on the second flip, I could get tails on the first flip , heads on the second flip, or I could get tails on both flips. So there's four distinct equally likely possibilities. Four distinct equally likely outcomes here. And one way to think about it is on the first flip I have 2 possibilities, on the second flip I have another 2 possibilities, I could have heads or tails, heads or tails, and so I have 4 possibilities for each of these possibilities, for each of these two I have 2 possibilities here. So either way I have 4 equally likely possibilities. And how many of those meet our constraints? Well we have it right over here, this one right over here having 2 heads meet our constraints. So this is ... and there's only one of those possibilities, I've only circled one of these four scenarios, so there is a one forth chance of that happening. Another way you could think about this, and this is because these are INDEPENDENT EVENTS and this is a very important idea to understand in probability. We'll also study scenarios that are not independent, but these are independent events. What happens in the first flip, in no way affects what happens in the second flip, and this is actually one thing that many people don't realize. There's something called "The Gambler's Fallacy" when someone thinks: "If I got a bunch of Heads in a row, then all of a sudden becomes more likely on the next flip to get a tails. That is not the case. Every flip is an independent event. What happenned in the past in these flips does not affect the probabilities going forward. So the probability of getting a head on the first flip in no way ... or the fact that you got a heads on the first flip, in no way affects that you got a heads on the second flip. So if you make that assumption, you could say that the probability of getting heads and heads, or heads and then heads is going to be the same thing as getting probability ... as the probability of getting heads on the first flip times the probability of getting heads on the second flip. And we know that the probability of getting heads on the first flip is 1/2, and the probability of getting heads on the second flip is 1/2, and so we have one half times one half, which is is equal to one forth, which is exactly what we got when we tried out all of the different scenarios, all of the equally likely possibilities. Let's take it up another notch. Let's figure out the probability ... and we kind of bit ignoring tails, so... let's pace on the tension to tails, the probability of getting tails, and then heads and then tails. So this exact series of events. I'm saying this exact order: the first flip is a tails, second flip is a heads and then third flip is a tail. Once again these are all independent events: the fact that I got tails on the first flip in no way affects the probability of getting a heads on the second flip and that in no way affects the probability of getting a tails on the third flip. So, because these are independent events we can say it is the same thing as the probability of getting tails on the first flip, times the probability of getting heads on the second flip, times the probability of getting tails on the third flip. And we know these are all independent events, so this right over here is one half times one half, times one half. 1/2 times 1/2 is one fourth, 1/4 times 1/2 is equal to one eighth. So this is equal to one eighth. And we can verify. Let's try all of the different scenarios again. So you could get heads, heads, heads, you could get heads, heads, tails, you could get heads, tails, heads, you could get heads, tails, tails, you can get tails, heads, heads, this is a little tricky sometimes, you want to make sure you're being exhaustive in all the different possibilities here, you could get tails, heads, tails, you could get tails, tails, heads, or you could get tails, tails, tails. And what we see here is that we got exactly eight equally likely possibilities We have eight equally likely possibilities, and the tail, head, tail is exactly one of them. It is this possibility right over here. So it is one of eight equally likely possibilities.