Thanks to Ed Pegg for reminding me of the somewhat more difficult
questions about BINGO.
The game of BINGO comes in various flavors. In the one I know, each
player is supplied with a 5x5 board. The first column is filled with
five different numbers drawn at random from {1,2,...,15}, the second
column from {16,...,30}, and so forth, except that the center square
is marked out. A random number generator supplies numbers from 1 to
75, and when a number appears on a board, the player marks out that
square. The first player to get five marks in a row vertically,
horizontally, or diagonally wins.
We assume that each board is equally likely, though that can't be
true--with over 552 septillion possible boards, only a small fraction
of them have ever been printed! Also I don't know whether the
bingo-printing industry prints new boards all the time. It's possible
they have a relatively small set of boards they print over and over.
Do any funsters know?
A second question: After N numbers have been drawn, what's the
probability you have five in a row? I don't have an easy answer, but
somehow I suspect that in the intersection of math funsters and
generating funcsters there should be one.
And a question that Ed passed on from Paul Stephens: Suppose we have a
side bet on which of the 12 lines will be filled by on the winning
board (and in a small number of cases, whether the winner has multiple
lines). What are the odds on this bet? How does it vary as a
function of the number of players?
There are at most five equivalence classes of lines: center
horizontal(1), off-center horizontal(4), center vertical(1),
off-center vertical(4), and diagonal(2). Is the distribution for
diagonals the same as for the center horizontal? I'm almost sure of
that, but somehow not satisfied. Is the board symmetry group of order
32, or larger? Does this interact in any interesting way with S_75?
Dan
Hoey(a)AIC.NRL.Navy.Mil
