Recreational Math

Card and Dice Games

Why isn’t it easy to win a powerball lottery
if there are so many numbers to choose from?

Don’t think powerball lotteries give a person an edge. For example, a power-ball lottery can be one in which 5 out of 50 balls (or numbers) are drawn, with an extra powerball pulled out of a different number of balls. This is not like figuring the odds for a “drawing 6 out of 50 balls” contest, but is actually two separate lotteries: one with 5 out of 50 balls and one with the powerball group.

The probability of matching the first five balls is determined as above. But the powerball is separate. For instance, say the powerball is taken from a group of 36, making the powerball group’s odds 36:1. The probability of winning the entire jackpot can be determined by adding this to the results of the five-ball draw. Now the odds become even higher: The 50/5 drawing + a powerball of 36 = (2,118,760 × 36):1 = 76,275,360:1 or about 76 million to one. And there are even worse odds if more balls are in the powerball group.

As for state powerball lotteries, there are plenty. According to the Multi-state Lottery Association, powerball lotteries are, as of this writing, played in 42 states, Washington, D.C., and the U.S. Virgin Islands. The list below gives the Multi-state Lottery Association’s own odds for winning the powerball, but from the calculations above, their numbers are a bit off, so don’t be fooled:

Match Prize Odds
5 balls + powerball Grand Prize 1 in 195,249,054.00
5 balls $200,000 1 in 5,138,133.00
4 balls + powerball $10,000 1 in 723,144.64
4 balls $100 1 in 19,030.12
3 balls + powerball $100 1 in 13,644.24
3 balls $7 1 in 359.06
2 balls + powerball $7 1 in 787.17
1 ball + powerball $4 1 in 123.48
powerball only $3 1 in 61.74


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