5. Suppose that you are offered the following deal. You roll a die. If you roll a 6, you win $10. If you roll a 4 or 5, you win $5. If you roll a 1, 2, or 3, you pay $6. Let X be a random variable that represents winnings. a. Construct the probability distribution for X. b. Over the long run of playing this game, what are your expected average winnings per game? c. Based on numerical values, should you take the deal? Explain your decision in complete sentences. 6. 10,000 tickets will be sold to a lottery for which the grand prize is $20,000. There are twenty additional prizes of $500. Each ticket costs $10. Let X be a random variable that represents the amount that is won after taking into account the cost of the ticket. a. Construct the probability distribution for X. b. Over the long run of playing this game, what are your expected average winnings per game? 7. Approximately 25% of students at a local high school participate in after-school sports all four years of high school. A group of four seniors is randomly chosen. Let X be a random variable that represents the number of seniors in the sample who have participated in sports all four years. a. Use the binomial probability formula to complete the probability distribution for X. X P(X) 0 0.316 0.422 IN ? 3 0.047 b. Which value of X is most likely? Which value of X is least likely? c. On average, how many seniors in the sample would you expect to have played sports all four years? In other words, what is the mean of X? d. What is the standard deviation of X? e. What is the probability that all four seniors in the sample played sports all four years? f. What is the probability that two or fewer seniors in the sample played sports all four years