Gopher5

Revised 2/6/08

In the Gopher5 Lottery five balls are chosen at random without replacement from a container containing 47 balls, numbered 1 to 47. To play, a person chooses five numbers. Payoffs are made depending on how many numbers match the player's choices, according to the following schedule for a $1 bet, and have the indicated probability of occurring: Gopher5 Odds

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

5 of 5 - - - - Jackpot- - - - - - -1 / 1,533,939 - - - -1 / 1,533,939

4 of 5 - - - - - $500- - - - - - 210 / 1,533,939 - - - -1 / 7,304.5

3 of 5 - - - - - -$15- - - - - 8,610 / 1,533,939 - - - -1 / 178.2

2 of 5 - - - - - - -$1- - - 114,800 / 1,533,939 - - - -1 / 13.4

Total probability of winning = 123,621 / 1,533,939, or about 1 / 12.4

1 of 5 - - - - - - -$0- - - 559,650 / 1,533,939 - - - -1 / 2.74

0 of 5 - - - - - - -$0- - - 850,668 / 1,533,939 - - - -1 / 1.80

For example, the probability for the case where one matches 3 of 5 numbers is computed as follows: From combinatorial theory, there are 47!/5!42! = 1,533,939 ways to choose 5 numbers from a group of 47. (It is reasonable to assume that when the drawing is held each of these 1,533,939 ways is equally likely to occur.) The number of favorable ways of matching exactly 3 out of one's 5 choices is (5!/3!2!)*(42!/2!40!) = 10*861 = 8,610. Therefore the probability of matching exactly 3 out of 5 is 8,610/1,533,939.

The minimum Jackpot is $100,000, paid in a lump sum. Note that if there is more than one winner the Jackpot is divided evenly among the winners.

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 1 in 1.5 million of winning, one's expected payoff on a $1 bet is only 22.75 cents:

(500*210 + 15*8,610 + 1*114,800 + 0*559,650 + 0*858,668) / 1,533,939 = .2275.

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