Where I left off in the last video we were saying, “Well I have two dice like you know, we’re just playing monopoly two six-sided dice and I want to say what is the probability that I get a seven?” So when I add up the two rolls in the dice what's the probability I get seven? So I drew this grid here and this grid essentially represents all of the outcomes I could get with the two dice where on the top row that’s the outcomes on dice one. I could get a one, a two, a three, a four, a five or six and similarly for dice two these are all the outcomes I could get.

So each of these squares represent a particular outcome of both guys for example, this square means that I got a six on dice one and a six on dice two. And of course what does that mean that they added up to, they’ve added up to 12, right? We could go through all of them. Essentially we could take the sum of dice one and dice two. We say what do they add up to? Well this is two, this is three, four, five, six, seven and then this will be three. It will go up, let’s see this will be three then this will be four, five, six, seven, and eight. This will be as to all them four and it’s keep going up five, six, seven, eight, nine. These were four plus one. This is five, six, seven, eight, nine, ten and I think you see a little interesting pattern here, right. This will be six, seven, eight, nine, ten, eleven and this is seven, eight, nine, ten, eleven, and twelve.

So if I said what's the probability of getting a seven? Well that’s all the squares that have a seven in them so let’s see. Let me see if I could use this fill two so this could be interesting. So where are all the sevens? This one, this one, this one, this one, this one and that one so what's the probability that I get the dice? It actually turned out pretty neat on that work.

What's the probability of getting seven? Well, from our one of the original definitions of probability we said, “Well, what are the total number of equally probable outcomes?” Well we have 36 outcomes and they're all equally probable, right. There are 36 total outcomes and so what's the probability we get seven? Well, how many of these 36 outcomes result in the dice adding up to seven? So six but the probability of getting a seven is equal to 6/36 is equal to 1/6. This grid is used for figuring out the probability of getting any number. We could say and we could even just by looking at this we see that the most likely of all of the numbers we get is seven, right.

If you just look at the pattern because it covers this whole diagonal in terms of and then the probability you're getting a six is equal to the probability of getting an eight. You know the probability of getting a nine is equal to the probability of getting the nine is equal to a probability of getting a five and so forth and so on. Actually let's do that, let’s say so seven is the most probable and just to get some intuition on dice rolls let’s say what's the second light so what's the probability going eight? And so how many eights are there out of the total number?

So the probability of getting an eight is equal to 5/36 and that’s also equal to the probability of getting a six, right, six of this probability of getting a six. So let me call the six is the same green so we know that that’s a six. So those were all sixes and you know this actually wouldn’t hurt to memorize because when you play monopoly you'll know your odds of landing on boardwalk for example. Well actually I probably do another video on what is expected value and expected cost and things like that because that’s probability plus a little bit of money and it will be very useful when you're playing monopoly.

So we can keep going what's the probability of five? There are four out of 36 outcomes or a five. The probability of a five is four out of 36 and that is equal to 1/9 and that’s also the same as probability of getting a nine so that’s interesting. I mean you know if you have ever play crafts or you play monopoly you now have a sense of what the different probabilities are of the different rolls and that’s why I think in a lot of game seven is a very important role because that is actually the most probable number. For example, the probability of getting a seven is higher than the probability of getting a nine or a five because what's the probability of five or nine that used of the or, well that’s the probability of getting five plus the probability of getting a nine which is equal to 1/9 + 1/9 which is equal to 2/9. Actually I was wrong. You see that’s why it’s good to use the calculation.

One sixth is less than 2/9 so this is the higher one but I can say so I was wrong about that but we can say that the probability of getting a two or an 11 is less than the probability of getting a seven. Let's calculate that. What's the probability of getting a two? There's only one situation where I get a two, so this is 1/36, one out of 36 results in a two and then the 11 that’s two out of 36, right. So two out of 36 and so that equals 3/36 and so that equals 1/12. So the probability of getting a two which is just this one or an 11 is one out of 12 so the probability of getting seven is actually twice that of getting a two or an eleven so that is just interesting I know. You know, sometimes I don’t know where this going but I think it’s interesting to analyze dice because dice show up a lot.

Another way although this grid is probably the clearest way of doing it, another way that I do it if I don’t have a grid in front of me, if I say what's the probability of getting a five? Well, it’s the probability of let’s say this is dice one and this is essentially the same thing as a grid but it’s good to have multiple frameworks of this. So how can I get a five? Well, if I get one on dice one I get a two, I get a four in dice two, I get a two then I need a three, if I get a three and I need a two, if I have a four I need a one and then if I have a five, those are the only situations.

So we could say what's the probability of getting so we need each of these probabilities and then the next one has to be this, so there are four probabilities that kind of keep us in the game in dice one. So what's the probability of getting a one? Oh, that’s 1/6 so this is dice one and so what's the probability of getting a one on dice one? That’s 1/6, right and there are 1/6 that’s probability you get a two, that’s probability you get a three, that’s probability you get a four, all right.

And so given that you got a one on dice one what's the probability to get a four then? So then there are six probabilities. You know, there is this three and actually you get a five or six but those won't count because we've be out of the game. So in dice one then on dice two there's a one out of six chance that I get a four and then there's you know a bunch of other chances following in numbers but this the only situation which we get a five.

And similarly on dice—this is dice two, this column and then if I get a two what do I need on dice two? I need a three. Well, to get exactly at there there's a 1/6 chance again and of course they sell up to five. I have a three here and then there's 1/6 chance that I get a two which is exactly what I need and of course there's a lot of other things we can get but we’re selecting for the fives.

If I need four there's a 1/6 chance that I get a one to get a five. So what are all the probabilities of this? Well, this is 1/6 times ones so the probability of this, of getting a one and then a four, so the probability of this event, of this one of getting a one and then getting a four. Well, that’s one of out of 36, 1/6 x 1/6 and then after that happens you have to get another 1/6 so that’s one out of 36 and by a similar logic this is one out of 36, this is one out of 36 and this is one out of 36. Each of these are one of out 36 and if you think about that grid we drew each of these outcome represent a square on that grid getting a two and then a three, getting a one and getting a four and then our total probability of getting five is a sum of all these four out of 36 which is equal to 1/9.

So I just want to show you, you don’t have to draw a grid. You could do a tree, you could do a little table like this and say what are the ways I can get five and what's the probability of each of these and then sum them up and they all work and in different times and different methods will be more useful. I will see you in the next video.