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4 can be chosen is 42 × C(49,5) = 80,089,128. We will need this number often and denote it by b (for big). The lottery officials go to great pains to make sure that all b possibilities are equally likely. So, a player has one chance in 80,089,128 of winning the jackpot. Of course, the player may have to share this prize. We note that the last column in Table 2 is labeled "odds" when it more properly describes the "probability of winning". Because the probabilities are small, there is not much difference between odds and probabilities. However, this is a good excuse to get the difference between the two concepts straightened out. The media prefers to use odds, and textbooks prefer to use probability or chance. Here the chance of winning the jackpot is 1 in 80,089,128, whereas the odds are 1 to 80,089,127 in favor (or 80,089,127 to 1 against). To win the $100,000 second prize, the player must get the 5 white numbers correct but miss the Powerball number. How many ways can this be accomplished? There is only one way to get the set of five white numbers, but the player's Powerball pick can be any of the 41 numbers different from the red number that was drawn. Thus, the chance of winning second prize is 41 in 80,089,128; rounded to the nearest integer this is 1 in 1,953,393. This is a good time to introduce the concept of independence. You could find the probability of winning the second prize by pointing out the probability that you get the 5 white numbers correct is 1/C(49,5). The chance of not getting the red ball correct is 41/42. Since these events are independent, the chance that they both happen is the product of their individual probabilities. We can also point out that the lottery numbers you pick are independent of those drawn to determine the winning numbers. On the other hand, your picks and those of other buyers cannot be assumed to be independent. Discussion Question: Why not? Prior to November 2, 1997, the Powerball game was conducted by drawing 5 white balls from a drum of 45 and one red powerball from a second drum of 45. The prize for getting the red ball correct was $1, and the ticket listed the chances as 1 in 84. This often seemed wrong to players who have had elementary probability as the following exchange from the Powerball FAQ* illustrates: * From the Multi-State Lottery Association web site at http://www.musl.com/PDF Image | USING LOTTERIES IN TEACHING A CHANCE COURSE
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