2. ['70 points] (gambling) To solve this problem I need to tell you about hypergeomem'c probabilities (we'll revisit this topic in the unit on discrete distributions) Suppose that you're considering a nite population of individuals, each of which can be classied in one of two ways (e.g., black and green balls in an urn, or Democrats and Republicans among people who stick to the major political parties). Let the total number of individuals in the population be N, of which N1 are of type 1 and N2 of type 2 (with N1 + N2 = N). If you now take a simple random sample (without replacement) of size n from this population, what's the probability that you'll end up with exactly 111 individuals of type 1 and ng of type 2? Evidently there are some restrictions here: 0 S m S N1, and 0 S n2 3 N2, and m + 712 = n. From our discussion of permutations and combinations, you can immediately see that there are ( N ) possible simple random samples, all of which are equally likely, and furthermore that there n N . . . N are ( n1 ) ways to choose the m type1 1nd1v1duals and ( n: 1 individuals of type 2. Thus ) ways to end up with exactly 712 Table 4: The nine ways to win in Powerbatt and the associated \"odds,\" as stated on the Powerbatt website. Match PI'lZe \"Odds\" \"$323.13? Grand Prize 1 in 29220133300 All ve Whites $1,000,000 1 in 11,688,053.52 Four whites . and the red $502000 1 m 913,129.18 Four whites $100 1 in 36,525.17 Three whites . and the red $100 1 m 14,494.11 Three whites $7 1 in 579.76 Two whites . and the red $7 1 1n 701.33 One white . and the red $4 1 m 91.98 The red $4 1 in 38.32 ( \"1 l ( N2 1 n1 n2 P(n1 type1 individuals and n2 type2 individuals) : _ (1) OK, now we can get on with the problem, which makes extensive use of these hypergeometric probabilities. Powerbatt 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 K12 education. In the Powerbatt game, ve numbered white balls are drawn in a manner certied 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 Powerbatt website. The rst error is that when the Powerbatt people said \"Odds\" in Table 4 what they really meant was \"the probability of occurrence, expressed as a fraction i.\" Another error was present in something the Powerbatt 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]. (a) Explain why the \"odds\" value in the rst row of Table 4 was not 1 in (69 - 68 - - - - -65 - 26) = 35,064,160,560, and why the stated \"odds\" value was essentially correct. [10 points] 5 11,238,513, and show that the lottery people got the correct answer. [10 points] 1 (b) Explain why the \"odds\" value for the Second Prize of $1,000,000 was not ( 69 ) = 1 in (c) For (1:: = 0, 1, . . . , 5), explain why the following formulas are correct: PG; whites and the red) I /'_'\\ 01% \\._./ /'"'\\ Hm \\__./\\--/\\__./ ( 5 ) ( 64 ) ( 1 k k PUC whites (and not the red)) 2 5 2: Use these formulas to verify the rest of the \"odds\" entries in Table 4. [30 points] (d) Show that the lottery people were right when they said that the overall \"0d \" of winning a prize are 1 in about 24.87, and explain why the statement \"The \"odds\" presented here are based on a $2 play (rounded to two decimal places)\" initially sounds ridiculous but can be made correct with the insertion of a single word. [10 points] (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]