Getting exactly two heads (combinatorics)

I'm going to start with a fair coin and I'm going to flip it I am going to flip it four times flip it four times and the first question I want to ask is what is the probability that I get exactly exactly one exactly one head or heads this is one of those confusing things when you're talking about the what side of the coin I know I've been not doing this consistently I'm tempted to say if you're saying one it feels like you should do the singular which would be head but I've write it up on the I ran up a little bit of it on the internet and it seems like when you're talking about coins you really should say one heads which is a little bit it seems a little bit difficult for me but I'll try to go with that so what is the probability of getting exactly one heads and I put that in quotes to say that well you know it's really we're just talking about one head there but it's called heads when you're dealing with coins anyway I think you get what I'm talking about and to think about this let's think about how many different possible ways we can get four flips of a coin so we're going to have one flip then another flip then another flip then another flip and this first flip has two possibilities it could be heads or tails the second flip has two possibilities it could be heads or tails the third flip has two possibilities it could be heads or tails and the fourth flip has two possibilities it could be heads or tails so you have two times two times two times two which is equal to 16 possibilities 16 16 possible outcomes when you flip when you flip a coin when you flip a coin four times sixteen possible outcomes and any one of the possible outcomes would be one of sixteen so if I wanted to say so if I were to just say the probability and I'm just going to not talk about this one heads if I just take a just you know maybe this thing that has three heads right here this exact sequence of events this is the first flip second flip third flip fourth flip getting exactly this this is exactly one out of a possible of sixteen events now with that out of the way let's think about how many possibilities how many of those 16 possibilities involve getting exactly one heads well we could list them you could get your heads so this is equal to the ability the probability of the probability of getting the heads in the first flip plus the probability of getting the heads in the second flip plus the probability of getting the heads in the third flip remember exactly one heads we're not saying at least one exactly one heads so probability in the third flip and then or the possibility that you get heads in the fourth flip tails heads and tails and we know already what the probability of each of these things are there are sixteen possible events and this is each of these are one of those 16 possible events so this this is going to be this is going to be 1 over 16 1 over 16 1 over 16 and 1 over 16 and so we're really saying the probability of getting exactly one heads is the same thing as the probability of getting heads in the first flip or the probability of getting heads or I should say the probability of getting heads in the first flip or the hit or heads in the second flip or heads in the third flip or heads in the fourth flip and we can add the probabilities of these different things because they are mutually exclusive both any two of these things cannot happen at the same time you have to pick one of these scenarios and so we can add the probabilities 1/16 plus 1/16 plus 1/16 plus 1/16 invested at 4 times we'll assume that I did and so you would get 4 16s 4 16s which is equal to 1/4 fair enough now let's ask a slightly more interesting question let's ask ourselves the probability probability of getting exactly two heads exactly two heads exactly two heads and there's there's a couple of ways we can think about it one is just in the traditional way and let's just do we know the number of possibilities and of those paths equally likely possibilities and we can only use this methodology because it's a fair coin so how many of the of the total possibilities have 2 heads of the total of equally likely possibilities so we know there are 16 equally likely possibilities how many of those have 2 heads so I've actually ahead of time so we save time I've drawn all of the 60 mean equally likely possibilities and how many of these involve two heads well let's see this one over here has two heads this one over here has two heads this one over here has two heads let's see that's this one over here has two heads and this one over here has two heads and then this one over here has two heads and I believe we are done after that so if we count them one two three four five six of the possibilities have exactly two heads so six of the 16 equally likely possibilities have two heads so we have a what is this a 3/8 chance of getting exactly two heads now that's kind of what we've been doing in the past but I want to do is think about in a way so we wouldn't have to write out all the possibilities and the reason why that's useful is we're only dealing with four flips now but if we were dealing with 10 flips there's no way that we could write out all the possibilities like this so I really want a different way of thinking about it and the different way of thinking about it is if we're saying exactly two heads you can imagine we're having the four flips flip one flip to flip three flip four so these are the these are the these are the flips or you could say the outcome of the flips and if you're going to have exactly two heads you could say well look I'm going to have one head in one of these positions and then one head and the other position so how many if I'm picking the first so I'm you know you could say and I'm going to you know I have kind of a heads one and I have a heads two and I don't want you to think that these are that these are somehow the heads in the first flip are the heads in the second flip what I'm saying is we need two heads we need a total of two heads in all of our flips and I'm just giving one of the heads a name and I'm giving the other head a name and what we're going to see in a few seconds is that we actually don't want to double count we don't want to count the situation we don't want to double count this situation heads one heads to tails tails and heads two heads one tails tails for our purposes these are the exact same outcomes so we don't want to double count that and we're going to we're going to have to account for that but if we just think about it generally how many different spots how many of different flips can that first head show up in well there's four different flips that that first head could show up in so there's four four possibilities for four flips or four places that it could show up in well if that first head takes up one of these four places let's just say that first head shows up on the first on the third flip then how many different places can that second head show up in well if that first head is in one of the four places and that second head can only be in three different places so that second head can only be I'm going to picking a nice color here can only be in three different can only be in three different places and so you know it could be in any one of these it could maybe be right over there any one of those three places and so when you think about it in terms of the first and I don't say the first head head one we actually let me call it this way let me call it head a and head B that way you won't think that I'm talking about the first flip or the second flip so this is head a and this right over there is head B so if you if you had a particular if I mean these heads are identical we these outcomes aren't different but the way we talk talk about it right now it looks like there's four places that we could get this head in and there's three places where we could get this head in and so if you multiply all of the different ways that you could get before all of the different scenarios where this is in four different places and then this is in one of the three leftover places you get twelve different scenarios you get twelve different scenarios different twelve different scenarios but there would only be twelve different scenarios if you viewed this as being different than this and let me rewrite it with our new so this is head a this is head B this is head B this is head a there would only be twelve different scenarios if you viewed these two things that's fundamentally different but we don't we're actually double counting because we can always swap these two heads and have the exact same outcome so what you want to do is actually divide it by two so you want to divide it by all of the different ways that you can swap two different things if we had three heads you would think about all of the different ways that you could swap three different things if you had four heads here would be all of the different ways you could swap four different things so you have there's twelve different scenarios if you couldn't swap them but you want to divide it by all of the different ways that you can swap two things so 12 divided by two is equal to six six different scenarios different fundamentally different scenarios considering that you can swap them scenarios if you assume that head a and head B can be interchangeable that it's it's it's a completely identical outcome for us because it's really they're really just heads so there's six different scenarios and we know that there's a total of we know that there's a total of sixteen equally likely scenarios so we could say that the probability of exactly two heads is six times six six scenarios and or we could there's a couple of ways you could say there are six scenarios that give us two heads of a possible 16 or you could say there are six possible scenarios and the probability of each of those scenarios is 1/16 but either way you'll get the same answer