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We are told that about 70% of tickets sold in a typical lottery are chosen by the easy pick method. Probably this percentage is even larger when the jackpot is large since people tend to buy a number of tickets and would be more likely to use the easy pick method when they do this. We will limit our calculation to the easy pick tickets which we estimate to be 70% of 210,800,000 or 147,560,000. The probability that a particular ticket is the winning ticket is 1/80,089,128. The probability of k winners can be obtained from a binomial distribution with p = 1/80,089,128 and n = 147,560,000. The expected number of winning tickets among those who choose the easy pick is this np = 147,560,000/80,089,128 = 1.84. Since p is small and n is large we can use the Poisson approximation: mk –m p(k ) = k ! e , k = 0,1,2,... where m = 1.84. Carrying out this calculation gives: No. Winners Probability 0 .159 1 .292 2 .269 3 .165 4 .076 5 .028 6 .009 7 .002 8 .001 90 From this we find that the probability, given that there is at least one winner, that the winner is the only winner = 292/(1-.159) = .347. Thus the probability that the winner has to share the prize is .653. Recall that the cash value of the July 29 jackpot was 161,500,000. Using the probability p(k) that we will have to share this with k others, we can find the expected amount that a player, with the winning numbers, will end up with. We need only sum the values (161,500,000)/k)*p(k) for k = 0 to 12. Carrying out this calculation we obtain the expected value of a jackpot winning $82,039,300. Now looking again at Table 5 we find that we still have a favorable game with expected value 1.2. 10PDF Image | USING LOTTERIES IN TEACHING A CHANCE COURSE
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