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How To Win At Bingo?


I bought a book called How To Win At Bingo, by Joseph E. Granville. The author says he can “increase the odds in your favor up to 50%.” I suppose he means that whatever your expected return is normally (certainly less than $1 for every dollar you invest in Bingo), that this book may improve this expected return to near 1.5 times that amount (probably over $1) In other words, he suggests that you will make a profit.

The idea is to choose your Bingo cards so that the numbers on the cards do not have bizarre, unlikely sequences on them. Examples:

     card A             card B
  B  I  N  G  O      B  I  N  G  O
  2 24 36 51 63      8 29 34 56 75
  3 18 39 49 64      5 24 31 46 69
  6 22 xx 50 66      1 23 xx 60 68
  7 16 31 47 61     12 16 45 59 62
  5 20 35 60 65     15 17 42 54 61

Card A has what the book calls “bad symmetry.” The numbers are mostly clustered around the low numbers for each column. Card B has “excellent symmetry.” The numbers are distributed much like the random distribution that you would expect from the random Bingo machine.

This all sounds reasonable, in a common sense kind of way. But it is complete foolishness, mathematically.

Every card has the same exact chances, as any other card. “Excellent symmetry” will not help you at all. A card that is all low numbers in order has the same winning chances as any other card:

  B  I  N  G  O
  1 16 31 46 61
  2 17 32 47 62
  3 18 xx 48 63
  4 19 33 49 64
  5 20 34 50 65

That is what mathematics says about Bingo.

Normally, I tend to pontificate, and wonder why people don’t believe me. Well, let me try to prove what I’m saying about Bingo:

Proof #1: To simplify the situation, let’s invent smaller B(ingo) cards:

 card X       card Y
    B            B
    1           12
    2            5
    3            7

We will choose numbers between 1 and 15, and 3 in a row wins. According to the spirit of the book, card X has “bad symmetry,” while card Y has “good symmetry.”

Which B(ingo) card is more likely to get the first hit? Mathematics says that every number is equally likely. The author of the book does not dispute this. 1 is as likely as 7. In fact the odds are 1/15 that any given number will be chosen on the first pick.

Well, then it must be the later picks which make X a bad card. For the purposes of this proof, let’s assume that we are tied with one hit each (3 & 12) after 4 picks.

 card X       card Y
    B            B
    1            X
    2            5
    X            7

The book might now argue that 2 or 1 are now not very likely. True. Very true. But 5 or 7 are also not likely. No combination of two specific numbers is very likely. In fact the chance of hitting a 1 (or a 5 or any other number) is now 1/11.

I can continue to argue that in all future situations (including when we are tied with two hits each), the actual numbers on the cards do not matter.

This proof (informal as it is) is valid. But it may not convince many people. Some people “know” that a 1 is not as likely as a 7, even though mathematics says it is. This is similar to the Gambler’s Fallacy. In both cases, a person’s hunches are more believable (to them) than actual reasoning.


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