(e) Suppose that T tickets were bought across the entire U.S. in a given week, that no one was clairvoyant or otherwise privy to knowledge about the winning numbers, and (for simplicity) that everybody made their lottery picks independently of everybody else. In the drawing on 30 Jul 2016, for which the Grand Prize (or jackpot) was $487 million, it could be estimated from historical records on numbers of tickets purchased as a function of jackpot size that T was about 182.9 million. Show that the chance of at least one Grand Prize winner on this occasion was about 46.5%. (In actuality, one winning ticket was sold in a supermarket in Raymond, New Hampshire.) [10 points] Table 4: The nine ways to win in Powerball and the associated "odds," as stated on the Powerball website. Match All five whites and the red All five whites Four whites and the red Four whites Three whites and the red Three whites Two whites and the red One white and the red The red Prize Grand Prize $1,000,000 $50,000 $100 $100 $7 $7 $4 $4 1 in 292,201,338.00 1 in 11,688,053.52 1 in 913,129.18 "Odds" 1 in 36,525.17 1 in 14,494.11 1 in 579.76 1 in 701.33 P(n type 1 individuals and n type 2 individuals) 1 in 91.98 1 in 38.32 = N N (96) N N (1) OK, now we can get on with the problem, which makes extensive use of these hypergeometric probabilities. Powerball is a national lottery in the U.S. with drawings every Wednesday and Saturday night at 10.59pm Eastern time. The money left over after paying the winners is used by each state for projects designated by the legislatures, such as helping to fund K-12 education. In the Powerball game, five numbered white balls are drawn in a manner certified by the lottery to be as close as humanly possible to at random without replacement from a drum containing white balls numbered from 1 to 69, and one red ball is then also drawn at random from a second smaller drum that has 26 numbered red balls in it. Table 4 lists the nine ways you can win and the "odds" against you. Each play of the game cost $2, and you can play as many times as you like. There are several errors on the Powerball website. The first error is that when the Powerball people said "Odds" in Table 4 what they really meant was "the probability of occurrence, expressed as a fraction "Another error was present in something the Powerball website further stated: The overall "odds" of winning a prize are 1 in 24.87. The "odds" presented here are based on a $2 play (rounded to two decimal places) [quotes added].