Answer to Problem of the Week for 08/24/98

Answer to August 24, 1998 Problem

Powerball Problem This summer, there was increased interest in the Powerball lottery when the jackpot offered $295 million. A group of machinists calling themselves "The Lucky 13" won the jackpot and took a cash payout of $161.5 million, with each receiving $8,944,447 after taxes. Their winning numbers were 8-39-43-45-49 and powerball 13. Powerball was first introduced in 1987 and the odds of winning were 1 in 55 million. Five white balls were picked from balls numbered from 1 to 45 and one powerball was picked from another set of balls numbered from 1 to 45. In November 1997, officials raised the odds of winning to one in 80 million, thus making likely jackpots of at least 100 million dollars twice a year. Now five white balls are chosen from those numbered from 1 to 49, and the powerball is chosen from another set numbered from 1 to 42. Suppose you want to lower the odds of winning Powerball to one in 300,000. What should the range of the white balls be and what should the range of the powerball be? (The maximum number of white balls that you may use is 50 and the maximum number for the Powerball is 50).
Solution to Problem: The odds are calculated by computing the combination of the white balls taken 5 at a time, and then multiplying by the number of balls from which the Powerball is chosen. For example, the odds for the original Powerball is figured in the following manner: 45 C 5 * 45 = 54,979,155 Likewise, the current Powerball odds: 49 C 5 * 42 = 80,089,128 After examining all combinations of balls up to 50, only one solution gives us odds of one in 300,000: Use white balls numbered from 1 to 18, and choose the Powerball from balls numbered 1 to 35. 18 C 5 * 35 = 299,880 The next closest values are: 19 White balls, 26 for powerball: 19 C 5 * 26 = 302,328 24 White balls, 7 for powerball: 24 C 5 * 7 = 297,528 23 White balls, 9 for powerball: 23 C 5 * 9 = 302,841

Powerball Problem

This summer, there was increased interest in the Powerball lottery when the jackpot offered $295 million. A group of machinists calling themselves "The Lucky 13" won the jackpot and took a cash payout of $161.5 million, with each receiving $8,944,447 after taxes. Their winning numbers were 8-39-43-45-49 and powerball 13.
Powerball was first introduced in 1987 and the odds of winning were 1 in 55 million. Five white balls were picked from balls numbered from 1 to 45 and one powerball was picked from another set of balls numbered from 1 to 45.
In November 1997, officials raised the odds of winning to one in 80 million, thus making likely jackpots of at least 100 million dollars twice a year.
Now five white balls are chosen from those numbered from 1 to 49, and the powerball is chosen from another set numbered from 1 to 42.

Suppose you want to lower the odds of winning Powerball to one in 300,000. What should the range of the white balls be and what should the range of the powerball be? (The maximum number of white balls that you may use is 50 and the maximum number for the Powerball is 50).

Solution to Problem:

The odds are calculated by computing the combination of the white balls taken 5 at a time, and then multiplying by the number of balls from which the Powerball is chosen.

For example, the odds for the original Powerball is figured in the following manner:
45 C 5 * 45 = 54,979,155
Likewise, the current Powerball odds:
49 C 5 * 42 = 80,089,128

After examining all combinations of balls up to 50, only one solution gives us odds of one in 300,000:
Use white balls numbered from 1 to 18, and choose the Powerball from balls numbered 1 to 35.
18 C 5 * 35 = 299,880

The next closest values are:
19 White balls, 26 for powerball: 19 C 5 * 26 = 302,328
24 White balls, 7 for powerball: 24 C 5 * 7 = 297,528
23 White balls, 9 for powerball: 23 C 5 * 9 = 302,841

Correctly solved by:

1. John McKay	Charlottesville, VA
2. Ryan Dutton	Winchester, VA
3. John Beavers	Winchester, VA

Answer to August 24, 1998 Problem

Correctly solved by: