Expert: Soroban - 11/28/2007

Question
Hi Soroban,
I person I met the other day runs a type of bingo on a cruise ship and I am wondering what the probability of winning this particular form of the game is: Participants are given a bingo card with 24 numbers on it.  To win, they must cross off ALL 24 numbers on the card.  40 numbers are drawn from a set of 75 balls numbered 1 through 75. Thanks.

Answer
Hello, Peter!

The game of Bingo does not have any easy answers.
I can give one VERY simplified version, though.

First of all, imagine that you are the ONLY one playing.

You have one set of 24 numbers on your card.
   It doesn't matter what set of numbers you have.
   You have exactly ONE set of numbers.

Forty numbers are drawn (from a set from 1 to 75).

What is the probability that YOUR set shows up?

There are:  C(75,40) possible sets of 40 numbers that could be called.

How many of them will contain YOUR 24 numbers?


Among the 40 numbers drawn, 24 of them must be yours.
    There is only ONE way to do that.
The other 16 numbers can be any of the other 51 numbers.
    There are:  C(51,16) ways.

Hence, there are:  C(51,16) ways you could win.


The probability is:  C(51,16) / C(75,40)

If you know about "combinations" and "factorials",
   you can crank out the answer.


And that's the best I can do for you.
If there is just one other player (in competition)
   the problem gets incredibly complicated.
.