Powerball

Powerball
Revised 2/6/08

In the popular Powerball Lottery five balls are chosen at random without replacement from a container containing 55 white balls, numbered 1 to 55, and then a sixth ball, the powerball, is chosen from a second container containing 42 red balls, numbered 1 to 42. To play, a person chooses five white numbers and a powerball number. Payoffs are made depending on how many white numbers match the player's choices and whether or not the powerball is matched, according to the following schedule for a $1 bet, and have the indicated probability of occurring: Powerball Odds

Matches - -Payoff- - - - - - - - - -Probability - - - - - Approximately

5W 1R - - - Jackpot - - - - - - - 1 / 146,107,962 - - - 1 / 146,107,962
5W 0R - - $200,000 - - - - - - 41 / 146,107,962 - - - 1 / 3,563,609
4W 1R - - - $10,000 - - - - - 250 / 146,107,962 - - - 1 / 584,432
4W 0R - - - - $100 - - - - 10,250 / 146,107,962 - - - 1 / 14,254
3W 1R - - - - $100 - - - - 12,250 / 146,107,962 - - - 1 / 11,927
3W 0R - - - - - $7 - - - - 502,250 / 146,107,962 - - - 1 / 290.9
2W 1R - - - - - $7 - - - - 196,000 / 146,107,962 - - - 1 / 745.4
1W 1R - - - - - $4 - - -1,151,500 / 146,107,962 - - - 1 / 126.9
0W 1R - - - - - $3 - - -2,118,760 / 146,107,962 - - - 1 / 69.0

Total chance of winning = 3,991,302 / 146,107,962, or about 1 / 36.6

2W 0R - - - - - $0 - - - 8,036,000 / 146,107,962 - - - 1 / 18.2
1W 0R - - - - - $0 - - 47,211,500 / 146,107,962 - - - 1 / 3.09
0W 0R - - - - - $0 - - 86,869,160 / 146,107,962 - - - 1 / 1.68

For example, the probability of getting 4W 0R is computed as follows: From combinatorial theory, there are 55!/5!50! = 3,478,761 ways to choose 5 numbers from a group of 55. There are 42 ways to choose 1 number from a group of 42. Therefore there are 3,478,761 × 42 = 146,107,962 ways of making a powerball selection. (It is reasonable to assume that when the drawing is held each of these 146,107,962 ways is equally likely to occur.) The number of favorable ways of getting exactly 4W out of one's 5 choices is (5!/4!1!) × (50!/1!49!) = 5 × 50 = 250; the number of favorable ways of getting exactly 0R out of one's 1 choice is 41. Therefore the total number of ways of getting exactly 4W 0R is 250 × 41 = 10,250, and thus the probability of getting 4W 0R is 10,250/146,107,962.

The minimum Jackpot is $15,000,000. However, this amount could be paid in 30 payments over a period of 29 years and so would not have a 'present value' nearly that large ($7,500,000 would be a reasonable estimate currently, and this amount could be taken out in a lump sum). Also, one must pay taxes on one's winnings, which would further decrease the amount one would actually win. There is also the possibility that more than one person would win, in which case the Jackpot would be split evenly among all winners* (this does not apply to any of the other prizes).
*See the movie, Bruce Almighty.

In general, one is less interested in the probability of winning than in how much on average one could 'expect' to win when one gambles. To compute the expected payoff on a bet one multiplies the amount of each payoff (prize) times the probability (odds) of getting that payoff, and adds these together. Note that if one ignores the Jackpot (See Reasonable Expectation), which one has a chance of less than one in 146 million of winning (one's chances of being hit by lightning are much greater), one's expected payoff on a $1 bet is only 19.7 cents!:

(200,000 × 41 + 10,000 × 250 + 100 × 10,250 + 100 × 12,250 + 7 × 502,250 + 7 × 196,000 + 4 × 1,151,500 + 3 × 2,118,760 + 0 × 8,036,000 + 0 × 47,211,500 + 0 × 86,869,160) / 146,107,962 = 28,800,030/146,107,962 = .197

One could increase one's expected payoff by using the Power Play option. For an additional $1 the expected payoff would be:

([400,000/4 + 600,000/4 + 800,000/4 + 1,000,000/4] × 41 + [20,000/4 + 30,000/4 + 40,000/4 + 50,000/4] × 250 + [200/4 + 300/4 + 400/4 + 500/4] × 10,205 + [200/4 + 300/4 + 400/4 + 500/4] × 12,250 + [14/4 + 21/4 + 28/4 + 35/4] × 502,250 + [14/4 + 21/4 + 28/4 + 35/4] × 196,000 + [8/4 + 12/4 + 16/4 + 20/4] × 1,151,500 + [6/4 + 9/4 + 12/4 + 15/4] × 2,118,760 + 0 × 142,116,660) / 146,107,962 = .690 (which is 3.5 × .197)

Since this costs $2, the expected return is 34.5 cents per dollar, which is still a very poor payoff.

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