2x + 3 = 5, solve for x. 2x + 3 = 5, solve for x 2x + 3 = 5, solve for x

1. (6 points) How large would a potential Jackpot have to be in the California Mega Millions Lottery in order for the game to be fair? Present your computation and explain your reasoning in addition to stating your final answer. Note: A fair game is one in which in the long-run neither party (lottery company and players) has an advantage. 2. There are a number of probability claims made in the Lottery tips. In this problem, you will verify whether these claims are true. Consider the current rules (5/70 + 1/25) of the CA MEGA millions game. (a) (5 points) Consider the Odd-Even Lotto Tip. Find the exact probability that the five white numbers selected will all be even. (b) (8 points) Consider the Sum Lotto Tip. Use the Central Limit Theorem to approximate the probability that the sum of the five white balls will fall between 132 and 223 (both inclusive). Discuss the appropriateness of using the Central Limit Theorem in this case. If any of the assumptions of the theorem are violated, state what these assumtions are and discuss how extreme (mild, moderate, severe) the violation is. (c) (6 points) Consider the Repeat Hits Lottery Tip. It is not entirely clear, whether this tip only concerns the white balls or also the gold MEGA ball. i. Find the exact probability that in a given drawing at least one of the white numbers is a repeat from the last drawing. ii. Find the exact probability that in a given drawing at least one of the six balls drawn (five white and one gold) is a repeat of the last drawing. (d) (5 points) Consider the Tip that recommends to avoid previous number combinations. How long, on average (in years), will it take until a given set of MEGA million lottery numbers (five white balls and MEGA ball) will be repeated in a drawing? Recall, that drawings take place bi-weekly and that assume that a year has 365 days (i.e., ignore leap years). 3. (5 points) Assume that game payouts are fixed (not pari-mutuel). Will following the advice of the lottery expert increase your chances of winning the lottery? (This is a multiple choice question - Yes/No answer ONLY.) 4. (10 points) Discuss the advice from the Lotto-Expert presented on page 2. Are the statements made in the individual tips presented there generally truthful? Do they hold for general (future) drawings, and not just for data collected in the past? Is following this advice a good idea when playing the lottery? Explain why or why not. Take a stand and defend your position with a good logical argument using the methods of probability you have learned in this course. Provide as much numerical evidence as necessary (you may cite your results from problem 2) to support your 3 Math 161A - Fall 2020 M. Bremer claim(s). Make sure to provide a cohesive argument (using correct spelling and good grammar). 5. (5 points) To study whether the drawing of lottery numbers for the MEGA million game is truly random, we will consider some data. The data (which are available in the file “Lottery Data.csv”) contains the date of the drawing, the five white balls drawn, and the MEGA ball drawn in all 1925 drawings of the NY MEGA millions between 5/17/2002 and 11/03/2020. Based on the graphs that you can find on the next pages, discuss whether it is reasonable to assume that lottery numbers are drawn independently and at random meaning that each possible number combination is equally likely to be drawn. Describe which aspect(s) of which graph(s) you base your conclusion on. Disclaimer: The data come from the New York State (not the California State) MEGA millions lottery. Like California, NY has changed the rules for the MEGA millions game several times over the last couple years. For instance, on October 19, 2013, they switched from drawing 55 white balls to drawing 75 white balls. On the same date, the number of possible MEGA balls were reduced from 45 to 15. Another rule change occurred in October 2017. Note: You are welcome to look at the raw data and/or create your own additional graphs if you wish, but that’s not required for this problem. Grading: Your work will be graded