The short version: Gambling is for dummies.

The long version:

On a trip to Vegas recently while waiting to enjoy a delicious breakfast at Cafe Bellagio (I highly recommend) I read this flier:

For the uninitiated, in this version of Keno you select 20 number (between 1 to 80) with no repeats (can’t select 12 twice for example) and write them down. Then, after you have selected all your numbers, 20 numbers are randomly chosen by the house (again between 1 and 80 no repeats). On a $5 bet the payout is listed above. If 0 of the numbers you selected are the same as the computers you get $500, if 1 is the same $10, 2 are the same $5, and so on. If all 20 numbers you select are the same as the computer you get a jackpot of $250,000.

I wondered what the odds were of winning the jackpot. Every number the computer selects has to be one of your numbers. The first one has a good chance (as you have all 20 numbers still) so it has a 20/80 chance of selecting a number you have selected. In the next selection it is a little harder, 1 number is already matched so you only have 19 numbers left, and the house only has 79 left to choose from so the chance is 19/79. Next time it is 18/78 and so on. If you write this down the formula, the total chance for getting all 20 is:

You can write this as

where k=20 and n=80 in our case. Some quick work on the calculator tells you the odds are

This is a staggeringly small number. To put this is perspective the age of the known universe is only 4.3*10^{17} seconds! In other words you would have had to play Keno 10 times a second for the entire life of the universe up until now to have a decent chance of winning. Needless to say the measly $250,000 you win won’t exactly be profitable.

While I suppose it shouldn’t come as news that you won’t make money gambling in Vegas, I was a bit taken back by how bad the odds were. In fact the most confusing thing about this is why the jackpot isn’t bigger, much much bigger. It is effectively impossible to win, so why not go all out: $10 million, $100 Million, $1 Billion! Think of the headlines that would make. More realistically why not at least $1 million, is has a nice ring to it, everyone wants to win $1 million, what do they have to lose? The only thing I can conclude is the people in charge don’t trust their statisticians.

The astute student of combinatorics (combinatorian?) will have recognized the above formula, it is simply the inverse of the “choose function”.

It describes how many ways you can choose n numbers out of k, or “n choose k” (80 choose 20).

It is also interesting to note that when calculating this, the scientific calculator on the iPhone will frequently overflow and give an error message, round off your number half way though, or even flat out give you the wrong result. If you calculate 80! then divided by 60! it gives 8.946*10^{36}. This is already incorrect, the correct answer is 8.601*10^{36}. If you then hit the “1/x” button to invert it, it gives 1.1*10^{-37}. This is actually correct (not overall but for the 8.946*10^{36} that we started with) even if it is rounded off dramatically. I realize that these are very large numbers and that they won’t fit into a float type (not sure why they didn’t use double, it is a scientific calculator after all) but it is a little disingenuous to report 13 significant digits on the first calculation (80!/60!) when there is really only 1 significant digit. The invert function is at least mostly honest with how (in)accurate it is (technically there should be some warning that your answer is rounded off).

So don’t play Keno, and don’t use your iPhone for rocket science.