1 solution.pdf - Stat 24400 Homework 1 Solution Total...

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Stat 24400 Homework 1 Solution Jan 12, 2017 Total points: 100 1. [20pts] The French Lottery (a) A Tirage is selected by choosing 5 numbers from 90. Therefore, the number of Tirages that can be selected is i. 90 5 = 43949268 . ii. Use Stirling’s formula, 90 5 1 p 2 · 90 1 - 5 90 - (90 - 5+1 / 2) 5 90 - (5+1 / 2) 44689346 . (b) In order to win your Terne bet, the lottery must choose your three numbers and two numbers from the remaining 87; for the Quaterne, your four numbers and one from the remaining 86. So if you pick k numbers, your probability of winning is p k = P (win with k # 0 s ) = ( k k )( 90 - k 5 - k ) ( 90 5 ) . You win $5 times the multiple with probability p k and $0 otherwise. Therefore the expected payo of the bets are Terne: p 3 · 5500 · $5 = $2 . 34 . Quaterne: p 4 · 75000 · $5 = $0 . 73 . Alternatively , the expected net gain are Terne: p 3 · (5500) · $5 - $5 = - $2 . 66 . Quaterne: p 4 · (75000) · $5 - $5 = - $4 . 27 . (c) From the hint, in order for the bet to be fair, the payo must equal 1 /P (win). We already calculated the probability of winning the Simple bets in part (b). Now we calculate the probabilities for the Determine bets. The Extrait is given in the hint. Without loss of generality, suppose that in the Ambe bet, you predicted your numbers would come in positions one and two in the Tirage. In order to win, the lottery must select your number in the first position, which has probability 1 / 90. Given that the first number matches, the probability that the second number matches is 1 / 89, so the probability of winning is 1 90 1 89 . The following table gives the resulting fair payo s. 1
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Type of Bet Fair Payo /Fair Net Gain Actual Payo Extrait Simple 18/17 15 Extrait Determine 90/89 70 Ambe Simple 400.5/399.5 270 Ambe Determine 8,010/8,009 5,100 Terne 11,748/11,747 5,500 Quaterne 511,038/511,037 75,000 Quine 43,949,268/43,949,267 1,000,000 (d) i. The event that you win at least one of the bets is the complement of the event that you win zero of the bets, meaning that you lose all of the bets. Therefore, the probability of winning at least one of the bets is 1 - P (lose all bets). In order to lose all of the bets, none of your numbers may appear in the Tirage.
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