Theoretical probability distribution example: tables | Probability & combinatorics

We're told that a board game has players roll two three-sided dice. These exist, and actually, I looked it up; they do exist and they're actually fascinating! And subtract the numbers showing on the faces. The game only looks at non-negative differences. For example, if a player rolls a 1 and a 3, the difference is 2.

Let d represent the difference in a given roll. Construct the theoretical probability distribution of d. So pause this video and see if you can have a go at that before we work through it together.

All right, now let's work through it together. So let's just think about all of the scenarios for the two dice. So let me draw a little table here. So let me do it like that, and let me do it like this. And then let me put a little divider right over here. For this top, this is going to be die 1, and then this is going to be die 2. Die 1 can take on 1, 2, or 3, and die 2 could be 1, 2, or 3.

And so let me finish making this a bit of a table here. What we want to do is look at the difference but the non-negative difference. So we'll always subtract the lower die from the higher die. So what's the difference here? Well, this is going to be zero if I roll a 1 and a 1.

Now what if I roll a 2 and a 1? Well, here the difference is going to be 2 minus 1, which is 1. Here the difference is 3 minus 1, which is 2. Now what about right over here? Well, here the higher die is 2, and the lower one is 1. Right over here, so two minus one is one, and two minus two is zero.

And now this is going to be the higher roll; die 1 is going to have the higher roll in this scenario. Three minus two is one, and then right over here, three minus one is two. Now if die 1 rolls a two, die 2 rolls a three, die three is higher; three minus two is one, and then three minus three is 0.

So we've come up with all of the scenarios, and we can see that we're either going to end up with a 0, a 1, or a 2 when we look at the positive difference. So there's a scenario of getting a 0, a 1, or a 2. Those are the different differences that we could actually get.

And so let's think about the probability of each of them. What's the probability that the difference is 0? Well, we can see that 1 out of the 9 equally likely outcomes results in a difference of zero. So it's going to be three out of nine, or one-third.

What about a difference of—let me use blue—1? Well, we could see there are one, two, three, four of the nine scenarios have that. So there is a four-ninth probability. And then, last but not least, a difference of 2? Well, there's two out of the nine scenarios that have that. So there is a two-ninths probability right over there.

And we're done! We've constructed the theoretical probability distribution of d.