Math 161 A - Writing Project
There are many lottery games in the United States. One of them is “MEGA Millions” offered by the State of California (and a number of other states). In this game, five of seventy white balls labeled {1, 2, . . . , 70} will be selected at random and without replacement in biweekly drawings. In addition, a gold “MEGA number” will be selected independently from the white balls from the set {1, 2, . . . , 25}. Players mark the five numbers and the mega number on tickets that they buy from a Lotto retailer. Each play costs $2. Assume that average payoffs in California are as follows:
Winning Combination Payout
All 5 of 5 and MEGA Jackpot
All 5 of 5 $ 1,000,000
Any 4 of 5 and MEGA $ 20,000
Any 4 of 5 $ 500
Any 3 of 5 and MEGA $ 200
Any 3 of 5 $ 10
Any 2 of 5 and MEGA $ 10
Any 1 of 5 and MEGA $ 4
None of 5 only MEGA $ 2
The payoffs are fixed in most states, except in California, where all prizes, including the Jackpot are pari-mutuel. This means that payouts depend on the number of ticket sales and the number of co-winners. For the purpose of this assignment, assume that the payouts for all prizes are fixed.On the internet, Lotto players can find many helpful hints from “Lotto-Experts” who suggest methods how a player can increase his or her chances of winning the lottery.Advice from one such expert can be found on the internet under http://www.smartluck.com/free-lottery-strategies.html
Select the California Mega Millions (5/70 + 1/25) game. Below you will find some of the advice provided there:
You can also find specific advice on how to select your lottery numbers on this website.
Three of these tips are shown below.
There also is advice on which number combinations should be avoided.
Among the specific advice for which combinations to avoid is this one:
Assignment: This assignment is worth 50 points which account for 10% of your course grade. Please work independently on this project and submit a well phrased solution (no longer than 5 single-spaced, typewritten pages in pdf format) on Canvas by 6pm on Monday 12/7/2020. You may use Latex or any other typesetting program for this project.
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 combi- nations. 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 3) to support your 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 rea- sonable to assume that lottery numbers are drawn independently and at random meaning that each possible number combination is equally likely to be drawn. De- scribe which aspect(s) of which graph(s) you base your conclusion on.
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