For asked answer questions about the game Plan CO. From the television game show. The price is right. So in this game, intestines have the opportunity to get chips to be dropped down onto a pegboard into slots labeled with cash amounts. We're told that every contestant is given one ship automatically and can earn up to four more chips by correctly guessing the prices of certain small items. Now, let P denote the probability a contestant correctly guesses the price of a prize. Then it follows that the number of chips they contestant earns they're invariable X can be modeled as X equals one plus n, where N is a random variable, which is by normally distributed with parameters for and N. In part, a were asked, determine the expected value of X and the variance of X. The number of chips a contestant earns. We have the expected value of X. Well, this is the same as the expected value of one plus end, which by linearity is the same as one, plus the expected value of N. And because an is by normally distributed, this is the same as one plus en, which is four times p Likewise, we have that the variance of X is the variance of one plus n. And so using property is a variance of a linear combination. This is one squared or one terms the variance of end because and is by noon really distributed. This is N, which is four times p times one minus p. Next, in part B, the ship amounts were given the amount of money one on the clink of board and were asked to determine the mean and variance of the winnings from a single chip. So we'll let w denote the winnings from one ship. We're going to use the probability mass function for W. So we have that the expected value of W This is the outcomes w could be. So we have the w could be anything from zero or 100 or 500 or 1000 or 10,000 and then time teach probability. So we have zero looking at the table times 0.39 plus looking at the table the outcome 100 times It's probability, which is three and so on up to the outcome 10,000 times its outcome 0.23 And after calculating we get. This is equal to $2598. Likewise, we have that the variance W well, this could be computed as the expected value of X squared minus the expected value of X. Sorry, W I mean, did I have w squared? How to find expected value of X w squared. We have the possible outcomes that w squared. So we have zero again times 00.39 plus 100 squared times point of three and so on. Up to 10,000 squared times Probability 10,000, which is 0.23 minus our calculation from before 2500 and 98 And this comes out to be 16 million 518,000 196 Next in part C were given a random variable y denoting the total winnings of a randomly selected contestant were given that the conditional mean and variance of the variable y Given that a player gets X chips arm you x and Sigma squared x respectively, where mu and sigma squared come from part B. We were asked to find expressions for the unconditional, mean and standard deviation of why Well, we have that by the law of total expectation unconditional, mean, expected value of why is equal to the expected value of the expected value of why given X, this is the expected value of well, we're given that the expected value of why given X is Mu X and so using linearity. This is the same as mu times the expected value of X which using calculations from before we have that new is 2000 598 and expected that you have x from part A was one plus four p. Likewise, we have that by the law of total variance, we have that the variance of why is the variance of the expected value of why given X plus expected value of few variants of why given X and from party we have that this is the variance of oh, sorry from this part were given that expected value of like given x again is mute X, and we're given that the variants of why given X is Sigma Squared X and by linearity for expectation by homogeneity for variance. This is mu squared variance of X plus sigma squared times, expected value of X substituting from previous problems you squared is 2598 squared times the variance of X, which is four p times one minus p plus Yeah, the Sigma squared is 2598. Or sorry, that's the wrong one. Sigma squared is 16,518,000 196 and the expected value of X was one plus four p. No simplifying and taking the square root. You get that. These standard deviation of why is thes square root of 165 million. We're sorry wants 16,518,000 196 plus 93 million 71,000 200 p minus 26 million 998,000 416 p squared. Finally, in Part D were asked to evaluate the answers from part C for P, equaling 0.5 and one well, using the formulas from Part C, we have that when P is equal to zero, the expected value of why is equal to 2598 times one plus four times zero, which is 2598 dollars, and we have that's thestreet nerd. Deviation of why, when P is equal to zero, this is going to be these square roots of 165 or sorry, 16,518,196. So this is simply 4000 and $64. So what is the interpretation of when P equals zero? This means that the contestant guesses incorrectly, always. So. If a contestant always guesses wrong, they we'll still get exactly one chip and the answers from part B apply. Now, let's consider when P is equal 2.5. Then we had the expected value of why, according to our formula, is 7794 dollars in the standard deviation of why is 7000 $504 Now? Let's consider when P is equal toe one. In this case, the expected value of why is 12,990 dollars, and the standard deviation of why is 9000 and $88 interpreting these last two results. In fact, all three results we see that as test instability to get chips improves. This is when P is increasing, then so does the contestants expected payout. We see that it increases from 2598 to 7794 up to 12,990. We see that the variability around that expectation also increases in a way we can reason about. This is because as the contestant has greater ability to get chips, they also have more opportunities. So mawr choices to make and therefore more vulnerability. Sorry, variability. But we have the standard deviation does not quite increase linearly with peace. So we see that it increases from 4000 to 7000 and then to 9000. So an increase of about 3000 500 between the first two and then only an increase of about 1000 500 for the second to