Probabilities and Progressive Bingo

 

            The purpose of this note is to estimate, via direct simulation on a computer, the probability of a player winning a round of progressive Bingo given the number of Bingo cards that are being played.

 

            In order to perform the calculations I wrote a computer program that directly simulates Bingo games using what is known as a Monte Carlo algorithm.  This is a numerical technique that was developed at Los Alamos during WWII for simulating the transport of neutrons through the fissionable material in an atomic bomb.  The Monte Carlo algorithm makes use of tables of random numbers synthetically generated on a computer.

 

            The procedure for simulating progressive Bingo is:

 

1.                  At the start of each game generate N synthetic Bingo cards.  These are full cards with 5 rows down and 5 columns across.  The center square, N3, is free.  The ranges of the numbers in each column are:

B:    1 – 15

I:   16 – 30

N:  31 – 45

G:  46 – 60

O:  61 – 75

 

The free square, N3, is denoted by zero.

 

An example Bingo card is shown in Figure 1.

 

Figure 1  Example BINGO Card

                  Generating such synthetic Bingo cards involves merely generating 5 random integers in the range 1 – 15 for the “B” column, 5 in the range 16 – 30 for the “I” column, and so on,

 

2.                  Next I generate a sequence of random integers in the range 1 – 75 that will represent the numbers drawn and called out during the game.  An example of the first 10 integers of such a sequence is:

 

15, 21, 73, 42, 6, 38, 13, 27, 64, 57

 

3.                  Now the game is played by picking the first number from the random sequence of 75 and cycling through the N cards looking for a match, then choosing the second random number from the sequence and again cycling through the cards, and so on.

 

4.                  If 51 draws in a game does not produce a Bingo another game is played with 52 draws.  If this does not produce a Bingo then another game is played with 53 draws being made.  This continues until a Bingo is achieved.

 

5.                  I call each sequence (1) – (4) an “experiment” and, in order to collect good statistics on the probability of obtaining a Bingo on a given number of draws with a given number N of cards out, I perform a very large number, typically 10,000 – 100,000 of such numerical “experiments”.

 

The results from the Monte Carlo simulations are shown in Figure 2.

 

Figure 2  Results from Monte Carlo simulation

 

The horizontal axis is the number of draws required in a sequence of Bingo games before a Bingo is recorded.  The vertical axis is the probability of achieving a Bingo in a given number of draws.  This probability obviously depends on how many cards are out.  I have chosen values for N, the number of cards out, of 200, 500, 1000, and 2000.  The number of numerical “experiments” performed for each value of N is 10,000 or 100,000.

 

As we would expect, for each value of N these probabilities have a peak or most likely value and fall off on either side of that.  The number of draws has an average value that decreases from 60 for 200 cards to 56 for 2000 cards.  Using 200 cards I did not observe a single Bingo in 51 draws in nearly a million games.  Using 2000 cards, however, the probability of obtaining a Bingo in 51 draws is about 1%.

 

The cumulative probabilities for 2000 cards are tabulated as follows:

 

            Number of Draws                     Cumulative Probability

51                                                                                        1%

52                                                                                        3.3%

53                                                                                        6%

54                                                                                        18%

55                                                                                        33.5%

56                                                                                        55.1%

57                                                                                        75.8%

58                                                                                        90.6%

59                                                                                        97.5%

60                                                                                        99.9%

61                                                                                        100%

 

This means, for example, that the total probability of scoring a Bingo in 55 or fewer draws is 33.5% or one in three (1/3).

 

            We can see what the true distribution of probability should look like (it is known as a multinomial distribution) by fitting a normal distribution – the so-called “bell curve”–through the calculated points.  Figure 3 shows the resulting smooth curve.

 

Figure 3  “Bell Curve” probability distribution

 

            The conclusions that I would draw from this analysis are developed as follows.  Given that the organization running the Bingo game has monetary costs associated with operating the game, the net revenues brought in, i.e. the total revenues minus what is paid out as winnings to the Bingo players, must exceed the operating costs in order for the game to be profitable.  Thus the hope is for the game to progress along for a number of games with the number of draws increasing by one with each successive game and with the “pot” increasing with each successive game.  As the house gets a certain percentage of the total revenue its take is maximized by having a long run of progressive games.

 

            Difficulties with this strategy arise, however.  If the number of players and/or cards is small the game may run out to 60 draws, say, but the revenues may be small.  If the number of players and/or cards is large more revenues may be taken in initially but the game will not typically progress as far because the probability of a Bingo becomes large for a smaller number of draws.

 

            In short, developing a strategy that is profitable for the house becomes an optimization problem that depends upon the price of the Bingo cards, the number of Bingo cards sold, the overhead associated with running the game, and the probability functions that I have calculated above.  As in any business, if the overhead is too high and the price of the Bingo cards is too low, the house will lose money.