In 2011, the Boston Globe reported that three highvolume betting groups, the MIT group, the Michigan group, and the Boston at Northeastern Universities group, have been monopolizing Cosh Winfall, a state lottery of Massachusetts. This report triggered an investigation on these groups as well as the Massachusetts State Lottery. There was no illegal activities found for all three groups and they each walked away with multi-million dollars, however, Cash Winfall was shut down after its last drawing on January 26, 2012 due to its aw in design. The goal of this project is to understand how these betting groups gamed the Massachusetts lottery system using probability theory. Cash Winfall was drawn on Mondays and Thursdays. Each ticket costed $2 and a player chose six distinct numbers from 1 to 46, ex: 1, 9, 1?, 23, 39, 40. To win the jackpot, these numbers must match all six of the numbered balls that drop out of the bin during a drawing. The minimum jackpot was $500,000 and could be higher depending on the lottery sales. The game also offered smaller prizes: $4,000 for matching ve out of six numbers; $150 for matching four out of six,' 55 for matching three of six; and a free 52 bet for matching two of the drawn numbers. There are two ways to pick six numbers. A player can handpick six distinct numbers out of 1416. Or, a lottery printing machine can be used to print a ticket. In the latter case, the lottery machine picks a random combination [all combinations are equally likely each time the computer prints a ticket}. The Massachusetts State Lottery {the Lottery in short] designed Cash WinFall to add 60 cents of every dollar wagered to thejackpot, leaving 40 cents for the Lottery to run its own operations and distribute to cities and towns in the form of local aid. Like all lottery games, Cash WinFall is designed to make money for the Lottery {and a net loss for the player] and the more Cash WinFall tickets the Lottery sold, the more money the Lottery made. Because the probability of matching all six numbers is very low, the jackpot was rarely claimed. In this case, like most lotteries, thejackpot was rolled over to the next period. In the very rare case of multiple winners, the jackpot was supposed to be split into the number ofwinners. However, in the history of Cash Winfall, that had never happened. However, Cash WinFall had a feature that distinguished it from other lotteries: the jackpot was capped at $2 million. If the jackpot reached $2 million and no one matched all six numbers, the money in thejackpot was distributed amongthe lower tier prizes for that drawing. The Lottery referred to this as a "roll-down" and it dramatically boosted payouts for lower tier prizes. Cash WinFall replaced Mass Millions, in game that lost players when it didn't produce a single jackpot winner in 2003. The Cash WinFall roll-downswere designed to generate excitement and ticket sales even if no one ever won thejackpot. and it worked. The rolldowns were a topic of conversation on onllne forums like lotterypost.com. The Lottery published the predicted amount of Jackpot of each game. The players flocked to Cash WinFall whenever the Lottery predicted a possibility of roll-down {over 52M jackpot}. Here's how a typical roll-down worked. The drawing held on Feb. 4, 2010 had a Jackpot of 51.631.236. No one matched all six numbers so the jackpot pool was carried over for the next drawing. By the next morning. the Lottery had estimated the Cash 1Il'sl'inFall jackpot for the Feb. 3 drawing at $2 million, meaning bettors could expect a roll-down provided no one matched all six numbers and claimed the entire jackpot. As was typical for drawings with anticipated roll-downs, wagering was heavy resulting $2.4M Jackpot by the drawing on Feb 3. And, guess what, no one hit thejackpot. As a result, that money rolled down to the next three lower prize levels using a formula developed by the Lottery when it created the game in 2004. Of the $2.4 million in the Feb. 3 Jackpot, 26 percent - about $633,333 went to boost the payout for those matching five out of six numbers. Instead of receiving the standard $4,000 prize, holders ofthe 35 tickets matching five out of six numbers received $22,035 - the 54,000 standard payout plus a 1f35th share of the $633,333 in roll-down money dedicated to this tier. Similarly, 4? percent of the $2.4 million about $1,153,000 was used to enhance the prize for those matching four out of six numbers. Holders ofthe 1,761 tickets matching four of six numbers received $307.52, instead of the standard payout of $150. For those matching three out ofslx numbers, about 2? percent of the Jackpot - $652,000 - rolled down to Increase the payout. Lottery records show there were 29,332 tickets matching three of six numbers. Each one was worth $26.35 Instead ofthe standard 55 prize. Lottery records also indicate thatJust over 203,000 tickets matched two of the six numbers drawn, entitling the holder to a free 52 bet {the prize stays the same for matching two of the drawn numbers}. Normally, a lottery is designed so that the expected payoff {prize - cost} is always negative {your expected prize money is less than the cost ofthe lottery]. However, the unique role of Cash WinFalI created an opportunity for the three betting groups mentioned earlier to rig the Cash WinFaII to obtain a positive expected payoff. 1. Suppose you purchased a single 52 Cash WinFall ticket. What's the probability of winning the rst place, second, third, fourth, and fth? What's the probability of winning nothing at all? 2. Suppose that 450,000 Cash WinFall tickets were sold for theJan 29th drawing. Six winning balls were drawn. What's the probability of having two winners of the jackpot? Assume all 450,000 tickets were printed by the lottery machine that randomly picks the numbers on each betting slip with equal probability. 3. Continued from #2, theJan 29\"\" drawing'sjackpot is announced to be $1,060,000 {$540,000 from new sales + $520,000 rolled over from the last drawing. Treating the 52 free betting as the equivalent dollar payoff, compute the expected payoff of your single 52 ticket. Is it worth investing in the lottery given the expected payoff