## Sunday, December 14, 2014

### Bingo: Number of Cards and Odds of Winning

Bingo

A standard BINGO board has a 5 x 5 matrix. Each column is dedicated to a letter, mainly B, I, N, G, and O. 75 numbers are assigned to each number as follows:

B: 1 to 15
I: 16 to 30
N: 31 to 45
G: 46 to 60
O: 61 to 75

Each column has 5 of 15 possible numbers, except the N column, which only has 4. The N column has a Free Space.

How Many Different Bingo Cards Are There?

A bingo card has five rows, five columns , and two diagonals. The middle row, middle column, and the two diagonals contain the Free Space.

Each of columns have the following number of arrangements:
B Column: nPr(15,5) = 15!/(15-5)! = 360,360
I Column: 360,360 (see B column)
N Column: nPr(15,4) = 15!/(15-4)! = 32,760 (remember the Free Space!)
G Column: 360,360 (like the B and I columns)
O Column: 360,360 (like the B, I, and G columns)

Note: nPr is the Permutation function n!/(n-r)!, sometimes labeled PERM or (n r) shown vertically. Simply put, permutation means arrangement: "How many ways can we arrange r out of n objects?

Since each of the permutations of B column can have each of the permutations of the I column; which in turn, each of those permutations can contain each permutation of the N column, and so on; the number of total number of bingo boards is:

360,360 * 360,360 * 32,760 * 360,360 * 360,360
= 360,360^4 * 32,760
= 552,446,474,061,128,648,601,600,000
≈ 5.52446 x 10^26

We can get the number of different cards by using rows as well.
1st Row: 15 * 15 * 15 * 15 * 15 = 15^5 =759,375 (15 numbers available per column)
2nd Row: 14 * 14 * 14 * 14 * 14 = 14^5 = 537,824 (14 numbers available per column)
3rd Row: 13 * 13 * 1 * 13 * 13 = 28,561 (don't forget about the Free space)
4th Row: 12 * 12 * 13 * 12 * 12 = 269,568
5th Row: 11 * 11 * 10 * 11 * 11 = 175,692

And the number of cards is:
759,375 * 537,824 * 28,561 * 269,568 * 175,692
= 552,446,474,061,128,648,601,600,000
≈ 5.52446 x 10^26

We arrived at the same destination.

What The Odds of Having a Winning Card in Bingo?

Even though there are 2,0711,126,800 ways to pick 5 numbers out of 75; thus making you very lucky to get a Bingo if the you win after just five draws.

Despite this, we look at the number of cards that are played in the game rather than the numbers themselves when answering this question.

So instead of having a long calculation, which I once thought was the case, the odds of having the winning card boils down to this:

1/(number of cards in play)

This is true if you are playing BINGO by a line or blackout bingo.

Don't forget to you yell BINGO! when you win. :)

Have a great day! Now I am off to errands and maybe wrap some presents.

Eddie

This blog is property of Edward Shore. 2014

P.S. I am going to make it one of my resolutions to check on the comments timely.  I appreciate and thank everyone who leaves a comment.   - Eddie (12/15/14)

1. Hi Eddie,

Love your blog. Been devouring it since I found it a few days ago.

I'm not sure about your calculation about number of ways to pick five numbers from 75. Shouldn't that be a combination nCr(75,5) = 17,259,390 rather than a permutation, because order doesn't matter in bingo?

--
Chris Randle

1. Chris,

Thanks - very much appreciated.

In this situation, I consider order important because of the order that the numbers are called during the game, helps determine who wins.

Eddie

2. What are the odds of winning Bingo in 4 and 5 numbers (1 in #)?

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