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birthday. We choose as the number of possible "birthdays" the number of possible first five numbers: C(45,5) = 1,221,759. We find that, for this number of possible birthdays and 17,001 people, the probability of finding 3 or more picks the same is .42. We also find that there is only a probability of .002 of finding 4 or more the same birthday. Thus we should not be surprised at finding 3 picks the same and should not expect to find 4 the same. Again, the computer picks seem to conform to random choices. Not surprisingly, we found no obvious patterns in the sets of numbers that were repeated more than once in the sets of numbers chosen by the computer. We look next at the 17,001 sets of 5 numbers chosen by the lottery players. We found 966 sets of numbers that were represented more than once (compared to 138 for the Easy Pick numbers). The largest number of times a particular set of numbers was chosen was 24. This occurred for the pick 02-14-18-21-39. Looking at the order in which the numbers were given to us, we noticed that these occurred consecutively in blocks of 5, with the blocks themselves close together. The ticket on which you mark your numbers allowed room for 5 sets of numbers. We concluded that one player had made 24 picks all with the same five numbers. He at least chose different Powerball numbers. The same explanation applied to the next most popular pick 08-12-24-25-27, which occurred 16 times. The third most popular set 03-13-23-33-43 was picked by 13 people and was more typical of the patterns that people chose. In this version of Powerball, the numbers were arranged on the ticket as shown below: Pick 5 EP__ 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 Note that the pick 03-13-23-33-43 is an arithmetic sequence obtained by going down a diagonal starting with 03. Similarly, the set of numbers 01-11-21-31-41, which was chosen 10 times, corresponds to going down a diagonal starting with 01 and the set 06-15-24-33-42, chosen 9 times, corresponds to going down a column starting with 06. The most interesting pattern noticed was 01-09-23-37-45, occurring 8 times, which results from choosing the corner points and the middle point. Since we do not expect repetitions of 4 or more to occur by chance, we looked at all those that occurred 4 or more times. We could explain all but three such sets of 5 numbers. These were: 16PDF Image | USING LOTTERIES IN TEACHING A CHANCE COURSE
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