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6 Number of ways n(1) = 1 n(2) = 41 n(3) = C(5,4) × C(44,1) n(4) = n(3) × 41 n(5) = C(5,3) × C(44,2) n(6) = n(5) × 41 n(7) = C(5,2) × C(44,3) n(8) = C(5,1) × C(44,4) n(9) = C(44,5) Match all six balls 5 white balls but not the red ball 4 white balls and the red ball 4 white balls but not the red ball 3 white balls and the red ball 3 white balls but not the red ball 2 white balls and the red ball 1 white ball and the red ball only the red ball Table 3: How many ways can you win a particular prize? Dividing these numbers by b, we obtain the chance of winning the corresponding prizes given in Table 2. Adding all the of n(i) values gives a total of 2,303,805 ways to win something. Thus we get an overall chance of winning of 2,303,805/b = 0.02877, which is about 1 in 35. In a textbook, we would be apt to give the results of Table 2 as: You Match 5 white balls and the red ball 5 white balls and not the red ball 4 white balls and the red ball 4 white balls and not the red ball 3 white balls and the red ball 3 white balls and not the red ball 2 white balls and the red ball 1 white ball and the red ball 0 white balls and the red ball You win JACKPOT $100,000 $5,000 $100 $100 $7 $7 $4 $3 Probability of Winning 0.000000012 0.000000511 0.000002746 0.000112624 0.000118118 0.004842854 0.001653657 0.008474995 0.013559992 Table 4: The probabilities of winning. As noted earlier, rounding the reciprocals of these probabilities to the nearest integer gives the numbers reported as "odds" on the lottery ticket. Discussion Question: Which of the two methods for presenting the chances of winning, Table 2 or Table 4, do you think is best understood by the general public? Which do you prefer? WHAT IS YOUR EXPECTED WINNING FOR A $1 TICKET?PDF Image | USING LOTTERIES IN TEACHING A CHANCE COURSE
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