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The value of a gambling game is usually expressed in terms of the player's expected winning. If there are n prizes and p(i) is the probability of winning the ith prize w(i), then your expected winning is: E = w(1)×p(1) + w(2)×p(2) + ... + w(n)×p(n) For all prizes, except the jackpot, we can assume we know the value of the prize. However, since the size of the jackpot differs significantly from drawing to drawing, we will want to find the expected winning for different jackpot sizes. In the 508 drawings from the beginning of the lottery on April 22, 1992 through March 1, 1997 the jackpot was won 75 times. It was shared with one other winner 11 times. During this period the jackpot prize varied from 2 million dollars to $111,240,463. If x is the amount of the jackpot and p(i) the probability of winning the ith prize, the expected winning is: E = x × p(1) + 100,000 × p(2) + 5000 × p(3) + 100 × p(4) + 100 × p(5) + 7 × p(6) + 7 × p(7) + 4 × p(8) + 3 × p(9) = x/b + 0.208 where b = 80,089,128. Using this, we can find the expected winning for various values of the jackpot. x = Jackpot ($ millions) 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 E = Expected Winning ($) 0.333 0.458 0.583 0.707 0.832 0.957 1.082 1.207 1.332 1.457 1.581 1.706 1.831 1.956 2.081 7 Table 5: Expected winning for different size jackpots. A game is said to be favorable if the expected winning is greater than the cost of playing. Here we compare with the $1 cost of buying a ticket. Looking at Table 5, we see that the lottery appears to be a favorable game as soon as x gets up to $70 million.PDF Image | USING LOTTERIES IN TEACHING A CHANCE COURSE
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