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 five flips of this fair coin and what I want to think about in this video is the probability of getting exactly 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 could be heads or tails second flip two possibilities there third flip two possibilities fourth flip two possibilities fifth lift two possibilities so it's 2 times 2 times 2 times 2 times 2 I hope I said that five times so - or you could view this two to the fifth power and that is going to be equal to 32 equally likely 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 we to figure out this probability we really just have to figure out how many of those possibilities how many of those possibilities involve getting three heads we could draw right out all of 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 5 flips here so let me draw the flips 1 2 3 4 5 and we want to have exactly three heads and I'm going to call those three heads I'm going to call those three heads I'm going to call them 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 disordering 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 different orderings if we cared about the difference between a B and C and then we're going to divide by that all of the different ways that you can arrange three different things so however many ways can we put a B and C into these five buckets that we can view as the flips if we care at about a B and C so let's start with a if 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 you know 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 if this was where the were heads a goes that 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 think if you if you cared about order how many different ways can you can you out of five different spaces allocate them to three different heads you would say it is five times four times three five times four is 20 times three is equal to 60 so you would say there's 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's aren't they are 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 then if this was heads be 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 kind of we don't have to over count for all of the different ways you arrange this and so 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 if I have three things that are in three spaces so for any so here I have a heads in the second in the second flip third flip and fifth lip 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 a can go into any of the three spaces 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 three times two times one way to arrange three different things so three times two times one is equal to six so the number of possibilities of getting three heads is actually going to be is actually going to be this five times four times three let me write this down in another color so the number of possibilities strike poss for short number of possibilities is equal to this five times four times three five times four times four times three times three over the number of ways the number of ways that I can rearrange three things because we don't want it we don't want to over count for all of these viewing the this this arrangement is fundamentally different than this arrangement so then we want to divide it by 3 times 2/3 I'm going to do that same orange color dividing it by 3 times 2 times 1 3 times 2 times 1 and which gives us in the numerator 120 120 divided by 6 so divided by 6 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 10 possibilities that gives us exactly 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 end of the 32 equally likely possibilities so you have a 5 over 16 chance of that happening