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Random numbers for experimental probability

AP Stats: UNC‑2 (EU), UNC‑2.A (LO), UNC‑2.A.4 (EK), UNC‑2.A.5 (EK), UNC‑2.A.6 (EK)

Video transcript

- [Instructor] Pascale Ricketts has invented a game called Three Rolls to 10. 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 10, you lose. What is the probability of winning Three Rolls to 10? So, there is 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 10 and figure out the proportion that we actually win and the more experiments we try, the better, the more likely that we're gonna get a good approximation of the actual probability. So, let's do that and to help us I'm going to have a computer generate a string of random digits from zero to nine. And the way that we're going to use this is remember, we're rolling a fair six-sided die, so the outcome could be one, two, three, four, five or six for each roll. In this random number list that the computer has generated, I do get digits from one to six but I also get the digits seven, eight, nine and zero and so what I'm going to do for each experiment, I'm going to start at the top left and I'm gonna consider each digit a roll. If it gives me an invalid result for a six-sided die, so if it's a zero, an eight, a seven or a nine 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, sum them up and we'll see how many we can do to figure out an experimental probability of winning Pascale's game. So, let me set up a little table here, so I want space to show the sum, so this is going to be the experiment, experiment, actually let me write the sum and over here we're gonna say did we win? Alright. So, let's start with experiment one. So, our first roll we got a one. Our second roll we got a five. We're doing quite well and then our third roll we got a six. Did we win? Well, one plus five plus six is 12. 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. So, 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 summed up to 12. So, we won. Alright, let's do another experiment. So, experiment number three. So, this first thing is invalid, so 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, that is invalid and then in our third roll we got a two, so we squeaked by. This adds up to 11. Yes, that looks like a win. Alright, let's do our fourth experiment here. So, our first roll we got a one. This is invalid. Second roll we got a two. This is invalid. Third roll we get a five. Did we win? One plus two plus five is eight. No, we did not win. So, that was our first non-win. So, let's keep going. This is interesting. Alright, this is invalid, so we're going to have, so this is trial five, we're going to have four plus three plus one. Four plus three plus one, adds up to eight. Did we win? No. Let's just keep going here. So, I'm gonna keep going with my table where I have experiment, I'll do five more trials, experiment, 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 'cause I wanna be able to look at our random numbers here. So, we are onto 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 10. 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, we will go to experiment eight. We get a one in our first roll, we get a three in our second row, this is invalid, the die fell off the table, we could 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 strong of non-wins. Alright. 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 got a five in our third roll. Did we win here? Yes, we won over here. This is definitely going to be greater than 10, this is 15 here. Alright, last experiment or at least for this video, last experiment. You could keep going. In fact, I encourage you to after this and 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 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 trials or 10 experiments, what is our experimental probability of winning this game? Well, out of the 10 experiments how many did we win? Well, it looks like we won one, two, three, four, five. So, based on just these 10 experiments we 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 instead of 10 times, maybe 10,000 times to see if we can get closer to the true theoretical probability.