Keno

Keno is a game where the player marks several numbers from the numbers 1 through 80.  Then the house randomly selects 20 numbers.  If many of the player's numbers are selected, then the player wins money.  The more selected the more money.  The game we will be talking about costs \$2 to play.  Below is a list of the payoff for each winning bet (Courtesy of Horizon Casino of Lake Tahoe).

Play 1 Spot

Catch 1  =  \$6

Play 2 Spots

Catch 2  =  \$24

Play 3 Spots

Catch 3  =  \$84
Catch 2  =  \$2

Play 4 Spots

Catch 4  =  \$240
Catch 3  =  \$6
Catch 2  =  \$2

Play 5 Spots

Catch 5  =  \$1500
Catch 4  =  \$22
Catch 3  =  \$2

Play 6 Spots

Catch 6  =  \$3000
Catch 5  =  \$200
Catch 4  =  \$6
Catch 3  =  \$2

Play 7 Spots

Catch 7  =  \$20500
Catch 6  =  \$750
Catch 5  =  \$32
Catch 4  =  \$2

Play 8 Spots

Catch 8  =  \$50000
Catch 7  =  \$3100
Catch 6  =  \$210
Catch 5  =  \$12

Play 9 Spots

Catch 9  =  \$50000
Catch 8  =  \$12500
Catch 7  =  \$900
Catch 6  =  \$50
Catch 5  =  \$4

Play 10 Spots

Catch 10  =  \$50000
Catch 9    =  \$17500
Catch 8    =  \$2200
Catch 7    =  \$320
Catch 6    =  \$32
Catch 5    =  \$2

Play 11 Spots

Catch 11  =  \$50000
Catch 10  =  \$50000
Catch 9    =  \$9000
Catch 8    =  \$1000
Catch 7    =  \$120
Catch 6    =  \$12

Play 12 Spots

Catch 12  =  \$50000
Catch 11  =  \$50000
Catch 10  =  \$16000
Catch 9    =  \$2500
Catch 8    =  \$420
Catch 7    =  \$44
Catch 6    =  \$10

We will focus on finding the probabilities for the Play 8 spots.  The rest are calculated in a similar manner.  First let us look at the probability of catching 5.  We can think of the eighty numbers as being divided into 20 good ones and 60 bad ones.  If we caught 5 out of 8 then five of the good ones are selected and 3 of the bad ones are selected.  There are

C(20,5) = 15504

ways of selecting 5 good ones and

C(60,3)  =  34220

ways of selecting the bad ones.  Hence the total number of ways of catching 5 is

C(20,5) x C(60,3)  =  (15504)(34220)  =  530546880

The total number of ways of selecting 8 numbers is

C(80,8)  =  28987537150

We get the probability by dividing

C(20,5) x C(60,3)
P(Catch 5)  =                                     =   0.0183
C(80,8)

We can compute the probabilities of catching 6, 7, and 8 in a similar way:

C(20,6) x C(60,2)
P(Catch 6)  =                                     =   0.00237
C(80,8)

C(20,7) x C(60,1)
P(Catch 7)  =                                     =   0.000160
C(80,8)

C(20,8) x C(60,0)
P(Catch 8)  =                                     =   0.00000435
C(80,8)

We can find the probability of losing the \$2 by subtracting from 1:

P(Lose)  =  1 - 0.0183 - 0.00237 - 0.000160 - 0.00000435  =  0.979

We can now write the probability distribution table:

 Winnings 10 208 3098 49998 -2 Probability 0.0183 0.00237 0.00016 4.35e-06 0.979

The expected value is given by

EV  =  (10)(0.0183) + (208)(0.00237) + (3098)(0.00016)
+ (49998)(0.00000435) + (-2)(0.979)

=  -0.79

We see that if you play many times, you will lose an average of 79 cents per game.

I will leave it as an exercise to figure out the rest of the expected values.

Always decide how much you are willing to spend before you begin.  Never go beyond this number.  If you find yourself pulling out more cash to get back what you lost, then you have a gambling problem and should call the National Council on Problem Gambling at 1-800-522-4700.