Keno : Keno game is a game with 80 numbers 1, 2, … , 80 where 20 numbered balls
ID: 2085496 • Letter: K
Question
Keno: Keno game is a game with 80 numbers 1, 2, … , 80 where 20 numbered balls out of these 80 numbers will be picked randomly. You can pick 4, 5, 6, or 12 numbers as shown in the attached Keno payoff / odds card.
When you pick 4 numbers, there is this 4-spot special that you place $2.00 in the bet, and you are paid $410 if all your 4 numbers are among the 20 numbers, or your are paid $4.00 if 3 of the 4 numbers 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 20 numbers 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…
(a)Based on this computation, is the payoff fair? Explain!
(b)The payoff for 3 numbers in your 4 chosen numbers appear in the 20 numbers is $4.00. Is that a fair payoff (how is the number of ways of 3 numbers matching related to the number of ways of 4 numbers matching?)?
Explanation / Answer
A player's entry can be any of C(80,3) combinations = 82160
From the the 20 numbers there are C(20,3) winning combinations = 1140
1140/82160 = .0138753 = Probability of having three winning numbers.
There are C(20,2) * 60 ways to get two winning numbers. 190*60 = 11400
(Two of 20 winning numbers, combined with any of 60 non-winning numbers.)
11400/82160 = .138753 = Probability of having two winning numbers.
There are 20 x C(60,2) ways to have one winning and two non-winning numbers: (there are 20 winning numbers and 60 non-winning numbers, of which we choose 2)
20 x C(60,2) = 35400 and 35400/82160 = .43087
There are simply C(60,3) ways to have no winning numbers = 34220
34220/82160 = .41650
Probability of losing your bet then is .43087 + .41650 = .84737
So your expected results are:
.0138753 * 41 + .138753 * 0 + .8473717 * -1 = -0.2784844
(3 winning numbers = +41, 2 winning numbers = break even,
otherwise -1)
For each dollar, you lose about 28c.
Making the game fair
There are many ways to make the game fair. In general, for a fair game,
we need 0.0138753 * (P + 10 * p) - .8473717 = 0.
where P is the three-number gain and p is the two-number gain.
In whole dollars, we could have
P = 61, p = 0, or P = 51, p = 1, or P = 41, p = 2.
Then the expected payoff (which would be the same in each case), would be:
(.0138753 * 61) + (.138753 * 0) + (.8473717 * (-1)) = -0.0009784
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