Random numbers for experimental probability | Probability | AP Statistics | Khan Academy

Pascale Rickets has invented a game called Three Rolls to Ten. You roll a fair six-sided die three times. If the sum of the rolls is 10 or greater, you win. If it is less than ten, you lose. What is the probability of winning Three Rolls to Ten?

So, there are several ways that you can approach this. The way we're going to tackle it in this video is we're going to try to come up with an experimental probability. We're going to do many experiments trying to win Three Rolls to Ten and figure out the proportion that we actually win. The more experiments we try, the better; the more likely that we're going to get a good approximation of the actual probability.

So let's do that! To help us, I'm going to have a computer generate a string of random digits from 0 to 9. The way that we're going to use this is: Remember we're rolling a fair six-sided die, so the outcome could be 1, 2, 3, 4, 5, or 6 for each roll. In this random number list that the computer has generated, I do get digits from 1 to 6, but I also get the digits 7, 8, 9 and 0.

What I'm going to do for each experiment: I'm going to start at the top left and I'm going to consider each digit a roll. If it gives me an invalid result for a six-sided die—so if it's a 0, 7, 8, or 9—I will just ignore that. I will just say, "Well that wasn't a valid roll." It's like you roll the die and it fell off the table or something like that.

So let's do that! Let's do multiple experiments of taking three rolls, summing them up, and we'll see how many we can do to figure out an experimental probability of winning Pascal's game.

So let me set up a little table here. I want space to show the sum; this is going to be the experiment. So let me write the sum, and over here we're going to say, "Did we win?"

All right, so let's start with experiment one. Our first roll, we got a one. Our second roll, we got a five. We're doing quite well! Our third roll, we got a six. Did we win? Well, one plus five plus six is twelve—yes, we won!

Let's do another experiment; this is going to be experiment two. We can just keep going here; these are random digits. We have a six in our first roll, we got a two in our second roll, we got a four in our third roll. Did we win? Yes, once again! This sums up to twelve, so we won!

All right, let's do another experiment. So experiment number three: the first thing is invalid. This is our first roll; we got a six, and then this is invalid. Our second roll, we get a three. This is invalid, that is invalid, and then in our third roll, we got a two. So we squeaked by; this adds up to eleven—yes, that looks like a win!

All right, let's do our fourth experiment here. Our first roll, we got a one—this is invalid. The second roll, we got a two—this is invalid. The third roll, we get a five. Did we win? One plus two plus five is eight—no, we did not win. So that's our first non-win.

So let's keep going; this is interesting! All right, this is invalid, so we're going to have—this is trial five. We are going to have 4 plus 3 plus 1. Four plus three plus one adds up to eight. Did we win? No!

Let's just keep going here. I'm going to keep going with my table where I have experiment, I'll do five more trials: x-bar, sum, and do we win? Let me make the table; this is just a continuation of the table we had before. I don't want to go below the page because I want to be able to look at our random numbers here.

So we are on to experiment six. Experiment six: we are getting a three in the first roll, a three in the second roll—this isn't looking good—and then a two in our third roll. Did we win? No, this is less than ten.

Now we go to experiment seven. Experiment seven: we get a two in our first roll, this is invalid. We get a three in our second roll plus three, and we get a one in our third roll. So plus one; once again, we did not win.

Now we go to experiment eight. We get a one in our first roll, we get a three in our second roll—this is invalid, the die fell off the table; we can think of it that way—and then in our third roll, we get a five. Plus five. Did we win? No, this adds up to nine.

So we had a string of wins to begin with, but now we're getting a string of non-wins. All right, now let's go to experiment nine. So we get a six in our first roll, we get a four in our second roll, and then these are all invalid, and then we get a five in our third roll. Did we win here? Yes, we won! Over here, this is definitely going to be greater than ten; this is fifteen.

All right, last experiment—or at least for this video. Last experiment. You could keep going; in fact, I encourage you to after this to see if you can get a more accurate, a better approximation of the theoretical probability of winning by doing more experiments to calculate an experimental probability.

So here, your experiment 10: first roll, we get a five; second roll, we get a two—this is invalid, invalid, invalid—then we get a six. Here we definitely won!

So with 10 trials, based on 10 experiments, what is our experimental probability of winning this game? Well, out of the 10 experiments, how many did we win? It looks like we won 1, 2, 3, 4, 5.

So based on just these 10 experiments, we've got a pretty clean 50%! So do you think the theoretical probability is actually 50%? Maybe you'd want to continue running these experiments over and over. Maybe we'd want to do a computer program that could run this experiment set of 10 times, maybe 10,000 times, to see if we can get closer to the true theoretical probability.