New York\'s \"Pick 10\" is a 10/80 lottery. Its payouts are set; they do not var
ID: 3173237 • Letter: N
Question
New York's "Pick 10" is a 10/80 lottery. Its payouts are set; they do not vary with sales. If you match all ten winning numbers, you win $500,000 (but you pay $1 to play, so your profit is $499, 999). If you match nine winning numbers, you win $6000. If you match eight, seven, or six you win $300, $40, or $10, respectively. If you match no winning numbers, you win $4. Otherwise, you lose your $1. a. Find the probabilities of winning first prize, second prize, and third prize. b. Use the results of pan a and the Complement Rule to find the probability of losing. c. Use the results of parts a and b to find the expected value of Pick 10.Explanation / Answer
Assuming that all the 80 numbers to choose from are unique, and let 'X' denote the number of matches.
P(X=10) = 10C10/80C10 = 6.07*10-13
P(X=9) = 10C9 * 70C1/80C10 = 4.25*10-10
P(X=8) = 10C8* 70C2/80C10 = 6.6*10-8
Similarly calculate for X = 7,6,5,4,3,2,1 and 0.
Now,
(a)
Probability of winning I prize = P(X=10)
Probability of winning II prize = P(X=9)
Probability of winning III prize = P(X=8)
(b)
P(losing) = 1 - P(winning)
P(winning) = P(X=10) + P(X=9) + P(X=8) + P(X=7) + P(X=6) + P(X=0)
(c)
E(Pick ten) = 10*P(X=10) + 9*P(X=9) + 8*P(X=8) + 7*P(X=7) + 6*P(X=6) + ... + 1*P(X=1) + 0*P(X=0)
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