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3 You Match You win 5 white balls and the red ball JACKPOT* 5 white balls but not the red ball $100,000 4 white balls and the red ball $5,000 4 white balls but not the red ball $100 3 white balls and the red ball $100 Odds 1 in 80,089,128 1 in 1,953,393 1 in 364,041 1 in 8879 1 in 8466 1 in 206 1 in 605 1 in 118 1 in 74 3 white balls but not the red ball $7 2 white balls and the red ball $7 1 white ball and the red ball $4 0 white balls and the red ball $3 Table 2: The chance of CALCULATING THE ODDS winning. The first question we ask is: how are these odds determined? This is a counting problem that requires that you understand one simple counting rule: if you can do one task in n ways and, for each of these, another task in m ways, the number of ways the two tasks can be done is n × m . A simple tree diagram makes this principle very clear. When you watch the numbers being drawn on television, you see that, as the five winning white balls come out of the drum, they are lined up in a row. The first ball could be any one of 49. For each of these possibilities the next ball could be any of 48, etc. Hence the number of possibilities for the way the five white balls can come out in the order drawn is 49 × 48 × 47 × 46 × 45 = 228,826,080. But to win a prize, the order of these 5 white balls does not count. Thus, for a particular set of 5 balls all possible orders are considered the same. Again by our counting principle, there are 5 × 4 × 3 × 2 × 1 = 120 possible orders. Thus, the number of possible sets of 5 white balls not counting order is 228,826,080/120 = 1,906,884. This is the familiar problem of choosing a set of 5 objects out of 49, and we denote this by C(49,5). Such numbers are called binomial coefficients. We can express our result as: C(49,5) = 49! = 49×48×47×46×45 5! 44! 5 × 4 × 3 × 2 × 1 Now for each pick of five white numbers there are 42 possibilities for the red Powerball, so the total number of ways the winning six numbers * The official Lottery explanation for the Jackpot is: "Select the cash option and receive the full cash amount in the prize pool. Select the annuity option and we will invest the money and pay the annuity amount to you over 25 annual payments." The cash payment is typically 50-60% of the total dollar amount paid over 25 years.PDF Image | USING LOTTERIES IN TEACHING A CHANCE COURSE
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