1. Suppose the powerball jackpot is reaching large numbers, and you' re thinking about buying a lottery ticket. As a rst cut, you're trying to decide if the expected value is larger than the cost of a ticket, which is $2. If the expected value is less than $2, you won't buy any tickets. (If the expected value is greater than $2, then you might want to use the Kelly criterion to decide how much of your wealth, if any, should be spent on lottery tickets.) Let's assume that you have a good idea of what the lump sum payout would be and that you have a good idea of the taxes that would be assessed on your winnings should you win. The biggest question is that if you should win, how many ways would you have to split the jackpot with others who purchased the same winning number. To get a handle on that, assume that we purchase one ticket and that ticket wins. Let X be the number of other people that purchased a ticket with the winning same number. I believe we showed that there are 292,201,338 different possible tickets. I heard on the radio that for one of the recent drawings where the jackpot was around 2 billion that 280 million tickets were sold. Let us assume that the same number of people buy a ticket in the next lottery and that each of those people purchase a ticket with the same number as yours with probability l/292,201,338. (This assumption might not be true. You could improve your expected winnings by selecting an unpopular number. It would be interesting to see which number are more popular and less popular. I'd guess that something like (l,2,3,4,5:1) would be a less popular number since it does not look random.) Let's also assume that the numbers purchased by different people are independent. (a) What would the support of X be? (b) What would be the distribution of X be including parameters? (c) As a function of X, What fraction of the jackpot would you receive assuming your ticket wins? (d) What other distribution including parameters would be an excellent approximation to the distribution of X? (e) Either compute or accurately approximate the expected fraction of the jackpot that you would receive assuming that you have a winning ticket