## 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.

Proof #2: Let's play real bingo, this time. You can have card B, while I will choose a made-up card C, chosen so that it has no numbers in common with card B. Card C is even more asymmetric than card A:

```     card C             card B
B  I  N  G  O      B  I  N  G  O
2 18 32 47 63      8 29 34 56 75
3 19 33 48 64      5 24 31 46 69
4 20 xx 49 65      1 23 xx 60 68
6 21 35 50 66     12 16 45 59 62
7 22 36 51 67     15 17 42 54 61```

But now, we are going to disguise all of the numbers in our Bingo game. A 2 becomes an 8, a 3 becomes a 5, etc., based on this table:

```  1 » 4    16 » 21    31 » 33    46 » 48    61 » 67
2 » 8    17 » 22    32 » 34    47 » 56    62 » 66
3 » 5    18 » 29    33 » 31    48 » 46    63 » 75
4 » 1    19 » 24    34 » 32    49 » 60    64 » 69
5 » 3    20 » 23    35 » 45    50 » 59    65 » 68
6 »12    21 » 16    36 » 42    51 » 54    66 » 62
7 »15    22 » 17    37 » 37    52 » 52    67 » 61
8 » 2    23 » 20    38 » 38    53 » 53    68 » 65
9 » 9    24 » 19    39 » 39    54 » 51    69 » 64
10 »10    25 » 25    40 » 40    55 » 55    70 » 70
11 »11    26 » 26    41 » 41    56 » 47    71 » 71
12 » 6    27 » 27    42 » 36    57 » 57    72 » 72
13 »13    28 » 28    43 » 43    58 » 58    73 » 73
14 »14    29 » 18    44 » 44    59 » 50    74 » 74
15 » 7    30 » 30    45 » 35    60 » 49    75 » 63```

We also use the same table to disguise all of our numbered balls which the machine will choose. Here are our disguised cards:

```     card C'            card B'
B  I  N  G  O      B  I  N  G  O
8 29 34 56 75      2 18 32 47 63
5 24 31 46 69      3 19 33 48 64
1 23 xx 60 68      4 20 xx 49 65
12 16 45 59 62      6 21 35 50 66
15 17 42 54 61      7 22 36 51 67```

As you can see, our disguised card B' looks just like our old card C. And C' looks like B. But they are not the old cards. Under our fake numbers are the old numbers. We are just using a code (a substitution cipher) for each number.

Well, by the definition of "symmetry" in the book, we find that C' now has "good symmetry" and B' now has "bad symmetry." And now, C' is much more likely to win (according to the book) than B'. That IS a contradiction. The book says that each card is both a better bet, and a worse bet.

Think about it. This IS a valid (but informal) proof.

Proof #3: For you die-hards, here's a third proof. I will play 10,000 games (on my computer) using only cards A and B, and see if one scores significantly better than the other.

OK, the results are in. After 10,000 games, card A won 4911 to 4865, with 224 ties. But, it's pretty even. Statistics shows that this data supports my hypothesis that the two cards have equal chances to win. And it does not support the alternative hypothesis that B is better than A. Incidentally, B should outscore A about half the time. This test definitely shoots down any 50% improvement in the odds, which the author claimed.

So, is there any way to make money at Bingo? Yes, indeed. Build a Bingo parlor (or a church), or write a book on How To Win At Bingo.