Using matrices to represent data: Payoffs | Matrices | Precalculus | Khan Academy

We're told Violet and Lennox play an elaborated version of rock-paper-scissors, where each combination of shape choices earns a different number of points for the winner.

So, rock-paper-scissors, the game, of course, where rock beats scissors, scissors beats paper, and paper beats rock. Then they give us this elaborate version right over here. When Violet wins, she gets two points. When Lennox wins with rock, she gets three. When Lennox wins with paper, she gets two points. When Lennox wins with scissors, she gets one point. If they choose the same shape, nobody gets any points because no one wins that round.

Complete the matrix so it represents their scoring system. It shows the number of points Violet gets; a negative number means Lennox gets those points. The rows are Violet's chosen shape, and the columns are Lennox's chosen shape.

So here we have the matrix right over here. I encourage you to pause this video and see if you can have a go at this on your own. If you have a piece of paper in front of you, I encourage you to get a piece of paper.

All right, now let's do this together.

So how many points? Remember, this matrix is how many points Violet gets, and if Lennox gets points, then it's a negative for Violet. So if Violet chooses rock and Lennox chooses rock, that is this entry right over here. What's going to happen? How many points is Violet going to get?

Well, if both players choose the same shape, nobody gets any points. So if they both get rock, rock will get a zero right there. We also know that's going to be true if Violet picks paper and Lennox picks paper; you're going to get a zero point for Violet there. And if they both pick scissors, that entry there will also give you a zero.

All right, now what if Violet picks rock and Lennox picks paper? What should I put there? Pause the video and think about it. Violet picks rock and Lennox picks paper.

Well, we know that paper beats rock, so this is a situation where Lennox wins with paper, and that corresponds to this scenario right over here. So Lennox will get two points. If Lennox gets two points, remember this matrix is all about what does Violet get. Violet gets negative two points right over here because Lennox got them.

All right, now what about this entry? What does that represent? Well, that represents Violet picking rock and Lennox picking scissors. We know that rock beats scissors. So this is a situation where Violet wins, and we know whenever Violet wins, she gets two points. So this will be a positive two points right over here.

Now what about this entry over here? Pause this video and think about what number goes there. Well, this is a situation where Violet picks paper and Lennox picks rock. We know that paper beats rock. So this is another situation where Violet wins, and she gets two points in any scenario where she wins. So that’s two points.

And now what about this one over here? Pause the video and think about what number goes there. Well, this is a situation where Violet picks paper and Lennox picks scissors. We know scissors beats paper because it can cut it up, I guess. Lennox has won with scissors; she gets one point.

So you might be tempted to write a one here, but remember that's Lennox getting a point. So this is all about how many points does Violet get? We said that would be negative one points if it's going to Lennox.

And then let's think about this last row here. What does this entry represent and what number should go there? Pause the video and think about it. Well, this is Violet picking scissors and Lennox picking rock. Now we know that rock beats scissors because I guess it can bash it up.

So Lennox in this scenario has won with rock, and we know that when Lennox wins with rock, she gets three points. So Lennox is getting three points here. This matrix is all about what does Violet get, so we want to put a negative three here. That’s three points for Lennox. Remember, a negative number means Lennox gets those points.

And one last entry, what do you think should go there? Well, this is Violet picking scissors and Lennox picking paper. We know that scissors beats paper because it can cut it up.

In any situation where Violet wins because she won with scissors here, she gets two points. So that is two points just like that.

So we've filled in the matrix. Now we have to answer this question: Assuming Lennox picks her shape entirely at random, what shape should Violet choose to maximize her chances of getting the most points? Pause the video and see if this matrix is helpful for figuring out the answer to that.

All right, so this obviously isn't an exercise on probability, but just as a little bit of a review, one way to think about it is when Violet picks rock, here are the scenarios; here are the outcomes that might happen.

Now they're telling us that Lennox picks at random. There would be a one-third chance that Lennox picks rock, one-third paper, one-third scissors. Since these are equally likely because they're saying that Lennox is picking at random, you can get what is sometimes known as the expected value here by taking the average of these three numbers.

Another way to think about it, it'll be one-third times zero plus one-third times negative two plus one-third times two. If you want to dig deeper into expected value, there's a lot of that on Khan Academy.

But we can really just take the average of these numbers, add them up and divide by three is another way to think about it. So here the expected value is going to be, if we take the sum, we get zero plus negative two plus two. Well, that all sums out to zero; divided by three, you get zero. So I’ll just write this zero here as the expected value when Violet picks rock and Lennox picks at random.

Now, in the second scenario, let's take the average. If we add all three of these up, you get two plus zero plus negative one, which is one; divided by three, you're going to get one-third as the expected value of the points for Violet.

Then in that last scenario, if you add all of these up, you get negative one; divided by three is negative one-third. So, it looks like the best expected value for Violet, assuming that Lennox is going to pick at random, is to go with paper. You have a positive one-third expected value.

So what shape should Violet choose to maximize her chances of getting the most points? Paper. Now, of course, that's assuming Lennox always picks at random. Obviously, if Lennox catches on that Violet keeps picking paper, Lennox would adjust their strategy, but that gets a little bit deeper.