DISCRETE MATH MATH 055
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Date Created: 10/22/15
Math 55 California Super Lottery April 13 1999 708 pm Every Wednesday and Saturday night the California Lottery stages a draw of six pingpong balls drawn at random from 51 balls numbered 1 2 3 51 Ticketholders who have paid 1 per ticket compare the six numbers drawn with the six numbers previously selected by the ticketholder or selected randomly by the Lottery at his request when he purchased the ticket A winning ticket s numbers must match at least three of the numbers drawn and the payback is then determined by the Lottery partially in accordance with the following schedule Table 1 Winning Tickets Prize Category Prize Value 6 out of 6 Jackpot 5 out of 6 Large Prize 4 out of 6 Medium Prize 3 out of 6 5 perticket Except for the 5 prizes each prize is divided equally among all holders of tickets in its prize category The total value of prizes awarded is required by statute to equal or exceed half the amount paid in for tickets except that if nobody wins the Jackpot it is carried over and added to the Jackpot for the next day s draw For example the draw on Wed 7 April 1999 had a Jackpot of 38000000 of which 25000000 was carried over from the previous few draws and the winnings were distributed as follows Table 2 Winnings for Wed 7 April 1999 Prize Category No of Winners Prize per Winner Total Paid 6 out of 6 2 19000000 38000000 5 out of 6 393 1523 598539 4 out of 6 22128 70 1548960 3 out of 6 406678 5 2033390 Totals 429201 4218088 9 Table 3 Winnings for Sat 10 April 1999 Prize Category No of Winners Prize per Winner Total Paid 6 out of 6 0 4000000 0 5 out of 6 82 1991 163262 4 out of 6 5066 83 420478 3 out of 6 97255 5 486275 Totals 102403 1070015 Since nobody won this Saturday s Jackpot it was carried over to the next Wednesday 14 April whose augmented Jackpot the Lottery has estimated to amount to 8000000 The Jackpot amounts are misleading because Jackpots are paid out in 26 annual installments unless the ticketholder has preselected a Cash Value option in which case the immediate payout is about half the stated value of the Jackpot depending upon bonds interest rates And then Federal and State taxes amounting to perhaps 40 of the actual payout are witheld Prof W Kahan Page 1 This r n nm an xxvna quotAnnl mm 17mm an Anlmr A n A Math 55 California Super Lottery April 13 1999 708 pm How are the odds calculated for this kind of gambling The number ofways to choose SlX balls is 51C6 18009460 Each ofthese ways is as likely as any other provided we assume rst that each ball is as likely to be chosen as any other and second that the choices are independent which means that after some balls have been chosen the next ball chosen is as likely to be any one of the remaining unchosen balls as any other of them These assumptions are essential to the presumption that the lottery is fair Only one of the ways of drawing SlX balls can match a ticketholder s choice so the odds against that ticket winning a share ofthe Jackpot are 51C6 18009460 to one For k 0 1 2 3 4 5 in how many ways can the draw match just k of a ticket s numbers That number of ways turns out to be 6Ck51 6C67k because there are 6Ck subsets of k out of Six numbers for k of the Six drawn balls to match and the other 67k drawn balls must be drawn from 5176 45 balls that match no other numbers on the ticket Then the probability that the drawn balls will match just k of the ticket s numbers is 6Ck51Ck51C6 and is tabulated below Table 4 Probability of a Draw Matching a Ticket s Numbers Match k out of 6 5Ck51 5C67k Ways Probability 1Odds Prize Value k 6 out of 6 1 555108 2 118107 Jackpot k 5 out of 6 270 15105 a 166702 Large Prize k 4 out of 6 14850 821041 211213 Small Prize k 3 out of 6 283800 0016 rs 1635 5 per ticket k 2 out of 6 2234925 k 1 out of 6 7330554 09834 a1716025 0 k 0 out of 6 8145060 Totals 5106 18009460 1 A Winning Ticket is one among whose SlX numbers are at least three matched by drawn balls In how many ways can the drawn balls do that At rst sight the number of ways may appear to be 6C351 3C3 345920 since there are 6C3 subsets of 3 out of 6 and the other three balls can be drawn from 5173 remaining balls But this kind of reasoning counts some of the ways more than once The correct count is 298921 obtained from either of 6C645C0 6C545C1 6C445C2 6C345C3 51C6 7 6CO45C6 7 6C145C5 7 6C245C4 which makes the odds against having a winning ticket 18009460298921 rs 6025 to one How much can ticketholders expect to win Since about half the money collected from them goes back as prizes we might think each 1 ticket worth about 050 on average But at times the Jackpot accumulates so much from previous losers that it seems to olTer a bargain as on Wed 7April then its 38000000 Jackpot might have made a 1 ticket with a 118009460 probability of winning the Jackpot seem to be worth at least 3 800000018009460 a 211 Alas bargains are not always what they seem Prof W Kahan Page 2 Math 55 California Super Lottery April 13 1999 708 pm First the 26year payout had an immediate cash value of about half the Jackpot s announced value second it had to be shared among two Jackpot winners This dropped the ticket s expected share of the Jackpot to about 053 Third taxes took about 40 of each share which further reduced the 211 to about 032 Perhaps it is unfair to take taxes into account that way since winners but not losers can subtract from their winnings the cost of their other losing tickets purchased the same year let s ignore taxes because tax laws are too complicated As we have seen the expected value of a ticket depends upon how many other tickets will share the bigger prizes and this depends upon how many tickets were purchased as well as the ways in which purchasers choose their numbers Some purchasers choose their numbers in nonrandom ways using birthdates or street addresses some purchasers distribute their choices more evenly than random numbers some purchasers choose numbers that have been drawn more often than others and some purchasers choose numbers that have been drawn less often The best we can do is to approximate these strategies by assuming that tickets are distributed as randomly as we think draws are distributed every subset of six numbers being as likely as every other And since the California Lottery publishes only how many tickets fell into each prize category not how many won nothing we shall have to estimate this number too Taking account of the previous Saturday s Jackpot of 25000000 Wednesday s needed about 13000000 from ticket sales to reach the announced value of 38000000 but these numbers are in ated by a factor of about 2 over actual cash payouts so the 13000000 really needed only about 6500000 cash to be paid from ticket sales The other prize payouts of 598539 1548960 and 2033390 are presumed to have been cash payments so all the prizes paid from ticket sales totaled roughly 10700000 Were this prize money half the amount realized from ticket sales the number of tickets purchased would have been about 21400000 but this is not quite consistent with estimates obtained in other ways The number of winning tickets 429201 is expected to be about number of tickets purchased odds against winning and these odds are about 6025 to one so the number of tickets purchased must have been near 4292016025 rs 25860000 by this reckoning Similarly 22128 tickets matched 4 out of 6 despite odds of 1213 to one so about 2212812l3 rs 26840000 tickets were purchased by this reckoning Now we have three estimates all contaminated by random noise and dubious hypotheses concerning the randomness of ticketholders choices Aplausible guess is that about 25000000 tickets were purchased How many of 25000000 tickets might reasonably be Expected to win the Jackpot This Expectation is 25000000 Odds against the Jackpot rs 2518 rs 139 which is not an integer Perhaps a better idea of what to expect is obtained from the estimated probabilities of nding k Jackpot winners among 25000000 tickets on the assumption that each ticket has independently a probability p 1 18009460 of winning the Jackpot From the textbook s theory of Bernoulli trials we nd those probabilities to be as follows for k 0 l 2 Table 5 Probability of k Jackpot Winners in 25000000 tickets k 25000000ckpk17p250000007k 0 025 l 035 2 024 3 011 Prof W Kahan Page 3
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