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Keno: Keno game is a game with 80 numbers 1, 2, … , 80 where 20...

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Basic Math

  1. Keno: Keno game is a game with 80 numbers 1,2, … , 80 where 20 numbered balls out of these 80 numbers will bepicked randomly. You can pick 4, 5, 6, or 12 numbers as shown inthe attached Keno payoff / odds card.

When you pick 4 numbers, there is this 4-spot special that youplace $2.00 in the bet, and you are paid $410 if all your 4 numbersare among the 20 numbers, or your are paid $4.00 if 3 of the 4numbers are among the 20 numbers chosen from the 80 numbers.

The number of ways of picking 20 numbers from 80 is C(80, 20) =80! / (60! 20!).

The number of ways that all your 4 numbers are among the 20numbers is: C(76, 16) (why?) = 76! / (60! 16!).

The probability that your 4 numbers bingo is C(76, 16)/ C(80, 20), and the theoretical payoff should be C(80, 20) / C(76,16) = 80! * 16! / (76! * 20!) = (80 * 79 * 78 * 77 ) / (20 * 19 *18 * 17 ) = $326.4355…

  1. (6%) Based on this computation, is the payoff fair?Explain!
  2. (6%) The payoff for 3 numbers in your 4 chosen numbers appearin the 20 numbers is $4.00. Is that a fair payoff (how is thenumber of ways of 3 numbers matching related to the number of waysof 4 numbers matching?)?

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