Probability using combinatorics
Exactly three heads in five flips
So let's start again with a fair coin. And this time, instead of flipping it four times, let's flip it five times. So five flips of this fair coin. And what I want to think about in this video is the probability of getting exactly three heads. And the way I'm going to think about it is, if you have five flips, how many different equally likely possibilities are there? So you're going to have the first flip-- let me draw it over here-- first flip. And there's two possibilities there. It could be heads or tails. Second flip, two possibilities there. Third flip, two possibilities. Fourth flip, two possibilities. Fifth flip, two possibilities. So it's 2 times 2 times 2 times 2 times 2. I hope I said that five times. Or you could view that as 2 to the fifth power. And that is going to be equal to 32 equally likely possibilities. 32-- 2 times 2 is 4, 4 times 2 is 8, 8 times 2 is 16, 16 times 2 is 32 possibilities. And to figure out this probability, we really just have to figure out how many of those possibilities involve getting three heads. We could write out all the 32 possibilities and literally just count the heads. But let's just use that other technique that we just started to explore in that last video. We have five flips here. So let me draw the flips-- one, two, three, four, five. And we want to have exactly three heads. And I'm going to call those three heads-- let me do it in pink-- heads A, heads B, heads C, just to give them a name. Although what we're going to see later in this video is that we don't want to differentiate between them. To us, it makes no difference if we get this ordering-- heads A, heads B, heads C, tails, tails. Or if we get this ordering-- heads A, heads C, or heads B, tails, tails. We can't count these as two different orderings. We can only count this as one. So what we're going to do is first come up with all of the different orderings if we cared about the difference between A, B, and C. And then we're going to divide by all of the different ways that you can arrange three different things. So how many ways can we put A, B, and C into these five buckets that we can view as the flips, if we cared about A, B, and C ? So let's start with A. If we haven't allocated any of these buckets to any of the heads yet, then we could say that A could be in five different buckets. So there's five possibilities where A could be. So let's just say that this is the one that it goes in, although it could be in any one of these five. But if this takes up one of the five, then how many different possibilities can this heads sit in? How many different possibilities are there? Well, then there's only going to be four buckets left. So then there's only four possibilities. And so if this was where heads A goes, then heads B could be in any of the other four. If heads A was in this first one, then heads B could have been in any of the four. I'll just do a particular example. Maybe heads B shows up right there. So once we've taken two of the slots, how many spaces do we have for heads C? Well, we only have three spaces left, then, for heads C. And so it could be in any of these three spaces. And just to show a particular example, it would look like that. And so if you cared about order, how many different ways can you, out of five different spaces, allocate them to three different heads? You would say it is 5 times 4 times 3. 5 times 4 is 20, times 3 is equal to 60. So you would say there are 60 different ways to arrange heads A, B, and C in five buckets, or five flips, or if these were people, in five chairs. And obviously, there aren't 60 possibilities of getting three heads. In fact, there's only 32 equally likely possibilities. And the reason why we got such a big number over here is that we are counting this scenario as being fundamentally different than if this was heads B, heads A, and then heads C over here. And what we need to do is say, well, these aren't different possibilities. We don't have to overcount for all of the different ways you arrange this. And so what we need to do is divide this by all of the different ways that you can arrange three things. So if I have three things that are in three spaces. So here I have a heads in the second flip, third flip, and fifth flip. If I have three things in three spaces like this, how many ways can I arrange them? And so if I have three spaces, how many ways can I arrange an A, B, and C in those three spaces? Well, A can go into three spaces. It can go into any of the three. Then B would have two spaces left, once A takes one of them. And then C would have one space left, once A and B take two of them. So there's 3 times 2 times 1 way to arrange three different things. So 3 times 2 times 1 is equal to 6. So the number of possibilities of getting three heads is actually going to be this 5 times 4 times 3. Let me write this down in another color. So the number of possibilities-- let's write "poss" for short-- is equal to this 5 times 4 times 3 over the number of ways that I can rearrange three things. Because we don't want to overcount for all of these, viewing this arrangement as fundamentally different than this arrangement. So then we want to divide it by-- I want to do that same orange color-- dividing it by 3 times 2 times 1. And which gives us, in the numerator, 120 divided by 6. Oh sorry, that's not-- it's 60 divided by 6. This is 60. 5 times 4 times 3 is 60. It gives us 60 divided by 6, which gives us 10 possibilities that gives us exactly three heads. And that's of 32 equally likely possibilities. So the probability of getting exactly three heads-- well, you get exactly three heads in 10 of the 32 equally likely possibilities. So you have a 5/16 chance of that happening.