Northstar Cash

Revised 2/6/08

In the Northstar Cash Lottery five balls are chosen at random without replacement from a container containing 31 balls, numbered 1 to 31. 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: Northstar Cash Odds

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

5 of 5 . . . . Jackpot . . . . . 1 / 169,911 - - - 1/169,911

4 of 5 . . . . . . $50 . . . . . 130 / 169,911 - - - 1/1,307

3 of 5 . . . . . . .$5 . . . . 3,250 / 169,911 - - - 1/52.3

2 of 5 . . . . . . .$1 . . . 26,000 / 169,911 - - - 1/6.54

Total probability of winning = 29,381 / 169,911, or about 1 / 5.78

1 of 5 . . . . . . .$0 . . . 74,750 / 169,911 - - - 1/2.27

0 of 5 . . . . . . .$0 . . . 65,780 / 169,911 - - - 1/2.58

For example, the probability for the case where one matches 3 of 5 numbers is computed as follows: From combinatorial theory, there are 31!/5!26! = 169,911 ways to choose 5 numbers from a group of 31. (It is reasonable to assume that when the drawing is held each of these 169,911 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!)*(26!/2!24!) = 10*325 = 3,250. Therefore the probability of matching exactly 3 out of 5 is 3,250/169,911.

The minimum Jackpot is $25,000, 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. If one ignores the Jackpot one's expected payoff on a $1 bet is 28.7 cents:

(50*130 + 5*3,250 + 1*26,000 + 0*74,750 + 0*65,780) / 169,911 = .287.

To figure the expected payoff including the Jackpot, one would add 'Jackpot/169,911' to the above. Assuming there is only one winner, this would add at least 25,000/169,911 = .147 for a total of .434 or more. (In order for the expected payoff to equal 1.000 (a fair game) the Jackpot would have to reach $121,161, again assuming there would be only one winner.)

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