 We're now going to talk about binomial distributions, means, standard deviation, which will then enable us to find the usual and unusual values of the distribution. So we have a game, and the probability that you win a game is 0.49. So the probability is 0.49. If you play the game 970 times, what is the most likely number of wins? So this is a binomial experiment because you have a fixed number of trials. The trials are independent, just because you win a game one time doesn't mean you're more likely to win it the next time or less likely to win it. And then the probability of a success, the probability of winning stays fixed. So my number of trials is 970, my probability of a success, meaning a success in this case is winning a game. So wins, a success is winning. And then Q is always 1 minus P, 1 minus 0.49, in this case 0.51. We will use this information now to calculate the mean and standard deviation. The mean, or to expect the number of wins if I was to play this game 970 times is number of trials times the probability of a success, 970 times 0.49, which is 475.3. So it would be expected that you would win 475.3 of those games, or if you want to think about it logically, 475. Find the standard deviation, so sigma is our standard deviation, and the formula is square root of N times P times Q, square root of 970 times 0.49 times 0.51. So that is square root of 242.403, which gives you 15.57. The standard deviation would be 15.57, or about 16 games. That is the mean and standard deviation of this binomial distribution, or binomial experiment. Now I want to go through and use the range rule of thumb to find the usual range of data values. Give your answers an interval using square brackets and only whole numbers. So I have my mean and standard deviation that I previously just calculated. So the range rule of thumb, it says that usual data values are within two standard deviations of the mean. So you take the mean, you add two standard deviations, you subtract two standard deviations. So minimum usual value would be mu minus 2 times sigma, that's 475.3 minus 2 times 15.57. Plug this into your calculator and you will get 444.16. So the minimum usual value of games that you would win would be 444.16. You have to be careful how you round this, because if I win 444 games, that is considered unusual because it falls below this minimum usual value. You will always round up, you will always round up for minimum usual value. Maximum usual value always round up. So 445 would be the minimum usual value. What about the maximum usual value? Maximum usual value, that is mu plus 2 times sigma, that is 275.3 plus 2 times 15.57. Which is 506.44. Now to win more than 506.44 games would be considered unusual. So cut off for a whole number of games, because you can't win part of a game, is 506. And this one would do it anyway, but in general for a maximum usual value, you should always round down. Regardless if this was 506.74, you would still round down to 506. So the interval for the usual number of games that you could win, usual values would be 445 comma 506. That is the interval for a usual number of games that you would win.