 One application where we have a sample space of equally likely outcomes concerns something called combinatorial probability. So again, let's consider the following experiment. We flip a coin ten times and we want to find the probability of obtaining exactly five heads in ten flips. And so check your intuition at the door. Do you think that this probability is fairly high or is this something that's somewhat unlikely? And we'll see whether your intuition matches with what our computation is. So we might make our sample space something like the number of heads, 0, 1, 2, 3, up to ten heads. The problem is that while this is a perfectly good sample space, your intuition should tell you that this sample space does not consist of equally likely outcomes. It does not seem that we are as likely to get say ten heads in ten flips as say five heads in ten flips. So your intuition should say that while this is a sample space, it doesn't consist of equally likely outcomes and we aren't going to be able to use it. So one way to change this is we might consider a different sample space which consists of the ways that the coin could land on each of our ten flips. So for example, just to keep everything organized, you might construct this table here and apologize for the very small print on the table. I couldn't fit it onto one line otherwise. But what we have is our first, second, third, all the way through our tenth flip. And what we're going to do is we're going to record what we actually get on each of those ten flips. And so maybe we get heads on the first and then heads on the second and tails on the third heads and so on. And we get our outcomes on each of the ten flips. And so if this is one outcome in our sample space, the thing we can start with is that each of the places here has two possible choices, the coin either lands heads or tails. And so that tells me that there's two to the tenth, one thousand twenty-four possible outcomes in our sample space. Now, are those outcomes equally likely? If we take the viewpoint that this outcome seems to depend on the probability of landing heads and or tails, and if we further take the assumption that those two outcomes are themselves equally likely, then this actual outcome here, heads, heads, tails, heads, tails, tails, heads, heads, seems to depend on the probability one-half, one-half, one-half, one-half, and all the way across. So this outcome here seems to depend on the probability one-half used ten times. And in fact, any other outcome that we can write would also seem to depend on the probability one-half used ten times. So it seems reasonable to conclude that the outcomes in our sample space do consist of equally likely outcomes because they seem to depend on the same probabilities used the same number of times. So it does seem that we have produced a sample space of equally likely outcomes. And so once we have our sample space of equally likely outcomes, the answer to the probability question then becomes a counting question. So if I want to see the probability of obtaining exactly five heads and ten flips, the question that I want to answer is how many of those outcomes have exactly five heads and ten flips? Well, one possibility is I could list all 1,024 outcomes and then look through that list of outcomes and then find which of those have exactly five heads. That would take a little while and probably lead to enormous eye strain. So let's see if we can find a more efficient way of doing that. And in cases like this, it turns out to be convenient to think of the problem as follows. While I have those ten outcomes, what I might do is I might choose five of those flips to be the flips that give me heads and the remaining five will be the flips that give me tails. And so in that way, I have exactly five heads with the remainder tails. So I can think about this as a set of choices. So the first head, the flip that will be the first chosen, well, there's any one of those ten flips could be that first head and then the second could be any one of nine, then eight, then seven and six. So here are the permutations that represent the ways that I can choose first flip to be head, second flip to be head, third flip to be head and so on. However, this is a permutation which means that whichever I pick first, second, third, fourth and fifth, I'm going to treat it as a distinct choice even if I choose the same flips but in a different order. So if I choose third flip, sixth flip, ninth flip, first flip, fifth flip, I would regard that as a different set of choices, first flip, second flip, third, sixth and ninth flips, even though it's the same things that are going to be heads. So there's an overcount here that I have to correct for. So let's see if we can correct for that. So the thing we might start with is the first flip we actually chose could be any one of the five that we did choose and the second could be any one of the four that we chose and so on down the line. And so on the top row, I have my permutations. On the bottom row, I have how many of those permutations correspond to a single choice of five flips and that'll give me my overcount factor. And so the actual number of ways that I have of choosing five flips to be heads is going to be product of the terms on the top row divided by product of the terms along the bottom row and that's going to be 252. Now, there's 1,024 elements in our sample space so the probability of getting five heads in 10 flips is going to be 252 out of 1,024, around 25%. Now go back to that intuition question which is did you think that the probability of getting exactly five heads would be high or would it be low? And here we see that it's about 25%. So it's not particularly low but at the same time it's not particularly high and this leads to a question that is worth thinking on for later. Because getting exactly five heads is not particularly likely about one chance in four, what would you conclude if you flipped a coin and didn't get exactly half the flips to be heads? Would you think that you were dealing with a fair coin or would you decide that you were not dealing with a fair coin? And that question is the heart and soul of statistics.