90 Number Bingo - Analysis

Introduction

Unlike American bingo, with a 5 by 5 card, with numbers from 1 to 75, in Europe and South America bingo is often played with a 3 by 9 card with numbers from 1 to 90. Below is an example.

As the example shows, the card contains 3 rows and 9 columns. On each row are exactly 5 numbers. The other four cells in each row are blank, or free squares. From other examples I have seen the first row contains the numbers 1 to 10, the second 11 to 20, and so on, but mathematically this doesn't matter. Winning events I have heard of all are based on covering rows only, so mathematically speaking the game could played on a 3 by 5 card with all numbers covered, the odds would be the same.

The following table shows the probability of covering 0 to 3 rows exactly by number of balls drawn.

Methodology -- Part 1

Following is how I did the math for the table above. First, let me define some variables.

• n = number of balls drawn.
• a = probability all three rows covered.
• b = probability at least two specific rows covered.
• c = probability at least one specific row covered.

Here are formulas for a, b, and c:

• a = combin(a,15)/combin(90,15)
• b = combin(a,10)/combin(90,10)
• c = combin(a,5)/combin(90,5)

Here are the formulas for exactly zero to three rows covered. For one and two rows, they can be any one or two.

• Exactly three rows covered = a.
• Exactly two rows covered = 3×(b-a).
• Exactly one row covered = 3×(c-2b+a).
• Exactly zero rows covered = 1 - (3c-3b+a).

Methodology -- Part 2

This section shows another way to get the probabilities in the table above.

The probability of covering m marks in c calls is combin(15,m)*combin(75,c-m)/combin(90,m). Using that, you can find the probability of covering a card as combin(75,90-m)/combin(90,m). To get the probability of covering 1 or 2 rows I determined the probability that m marks would cover 1 or 2 rows. The chart below shows those probabilities, which is based on basic probability.

 

The Wizard's Information About Bingo

• Bingo Probabilities
• Coast Casinos bingo
• Analysis of the Jumbo Progressive at Station Casinos
• Analysis of 90-Number Bingo

Written by:Michael Shackleford