A game that has become very popular in areas such as Las Vegas. So popular, in fact, that some casinos devote a channel to the results of the Keno game. The typical game has players pick twenty numbers between one and eighty in an attempt to pick the same twenty numbers a computer did. Consequently, the more numbers the player and the computer match, the more money the player wins. A typical list of payouts might look something like this:
Match---------Pays
20…………$100,000
19…………$100,000
18…………$100,000
17…………$100,000
16…………..$50,000
15…………..$25,000
14…………..$12,500
13……………$5,000
12……………$1,000
11……………...$200
10……………….$50
9……………...…$25
8……………...…$10
7………………….$5
6………………….$0
5………………….$0
4………………….$0
3………………….$5
2………………….$5
1………………...$10
0……………… $500
The game is deceptively easy and the list of winnings makes the game seem as if it is in favor of the player. However, as with most games of this type, the casino holds the better odds. Assume, for the purposes of this little demonstration, that the previous list is the list of payments and that tickets are $5 each. The total number of combinations of the twenty numbers that could be picked is 3,535,316,142,212,174,320, which is obtained by the equation 80! / (20! * 60!). Another way to state this equation is eighty numbers taken twenty at a time, but that really isn’t important unless you need to know the technical terminology. An easier method of getting this value is the combination function on most graphing calculators. On a TI-92, simply type in nCr (80,20) and it will give the correct answer.
To get the probability that the player and the computer will match a certain number of numbers, find the probability of matching that many multiplied by the probability of not matching the rest. For example, to find the probability to match twenty it is nCr (20,20) * nCr (60,0). The player wants to match all twenty numbers the computer picked, but none of the numbers it did not pick. Written longhand, the equation would be 20! / 20!, which equals one. Thus, the actual probability is 1 / 3,535,316,142,212,174,320. To match nineteen, it would be nCr (20,19) * nCr (60,1) or (20! / (19! * 1!)) * (60! / (1! * 59!)). This gives an answer of 1200, which is then divided by the total number of possibilities in order to get the probability. The process can be continued until all values are found for each of the twenty numbers.
After all the probabilities are found for the twenty numbers, the probabilities of winning, losing, and breaking even can be determined by adding the values together. To find the chances of breaking even, add the probabilities of anything that pays the amount of the ticket, in this case $5. By adding the chances of matching seven, three, and two, it is found that the probability of breaking even is approximately .28787, or 28.787%. By the same method, the probability of losing is 62.8487% and the probability of winning is 8.3694%.
With these values, it can also be determined, on average, what the casino will pay to the players on each ticket purchased. Simply take the probability of a number, for example, 18 (which has a probability of 336,300 / 3,535,316,142,212,174,320), times the amount paid if a player would match that many numbers, $100,000 in this case, to come up with the answer, 420,385,000 / 44,191,451,777,652,179. Do this for all the numbers and add them together. In this particular scenario, the casino would expect to pay approximately $3.54 on each ticket purchased, but since tickets are $5, the casino still comes up smiling. Moral of this little story: the casinos have built their buildings using money they have collected from all their players, so be careful before placing a bet on the table.