(Math notation is generally the same as
that used in Microsoft’s Excel. The
MathNotation link
will also give examples of the notnotation as used here.)
For 90 number Bingo ( 3 rows, 9 columns) please see
Bingo 90.
The
Bingo
Statistics link gives tables and graphs showing the probability for
getting “Bingo” after the announcer has called
“N” numbers. Tables and
graphs cover both a single board and a 50 board game.
The
Bingo 4
Corners and Letter “X” (both diagonals) link shows the
single board
probability of getting these patterns after “N” numbers
have been
called.
The
Bingo
Picture Frame (all 4 edges) and Letter “Y” link shows
the single
board probability of getting these patterns after “N”
numbers have been
called.
The
“How to
calculate” link shows how to calculate these numbers
including how
to calculate the probabilities when any arbitrary number of boards are
being played in a game.
Probabilities
for Swedish Bingo. A Swedish Bingo card has the familiar 5 rows and
5 columns, but the middle cell is not free. It has to be filled by
having its number called.
Rules of the game: A typical Bingo card has 24
semi-random numbers and a central star arranged in a square of 5 rows
and 5 columns. A Bingo card might look like:
1 16 31
46 61
4 19 34
49 64
8 23 *
53 68
11
26 41 56 71
15
30 45 60 75
We used the phrase “semi-random” to describe
the numbers
because the numbers in each column are confined within ranges. Column 1
will contain 5 random numbers in random order, but they are within a
range of 1-15. Similar ranges exist for the other 4 columns (16-30,
31-45, 46-60, and 61-75). The central location is a “Free”
spot. There
are (15!/10!)^4 * (15!/11!) = 5.52+ E26 (more than 552 million billion
billion) possible combinations that could exist - any one of which
would be a legal Bingo card. (The “!” symbol is the
mathematical
notation for Factorial. e.g. Factorial(5) = 5 * 4 * 3 * 2 * 1 = 120.)
Note: The above number of combinations assumes the numbers in any
column can be in random order. If the numbers in any column are always
in sorted order with the lowest number on row 1 and the highest number
on row 5, then the number of combinations in each of 4 columns is
reduced by 5! = 120 and the number of combinations in the center column
is reduced by 4! = 24.
Initially, the central “*” is counted as a
“free” or
“called” cell. Then, an announcer will call out numbers
selected
randomly within the total 1-75 range. (Usually this is done by randomly
removing numbered balls from a revolving drum.) Whenever one of
these called numbers matches a number on a player’s Bingo card,
the
player marks that number as “called”. Eventually, there
will be a
straight line of 5 called numbers that fill a row, fill a column, or
form a corner-to-corner diagonal line. (Note: the “Free”
center space
can be part of the straight line). At this point the player yells
“Bingo” and the game is over.
Probabilities: Of interest, in a single board
game - What is the probability the player will have a
“Bingo” after the
announcer has called “N” numbers? Also, in a multiboard
game, what is
the probability that the first “Bingo” will show up after
“N” numbers
have been called? (Check the
Bingo Statistics
link.)
Variations on Bingo: Other patterns can be used
for the game of Bingo. For example, a winning Bingo could be defined as
filling a 2x2 block anywhere on a Bingo card. There are 16 possible
locations where a 2x2 block could be located. Other Bingo variations
could include filling any of the 9 possible 3x3 blocks, or filling a
2x3 block. A 2x3 block could also be rotated for 24 possible winning
“Bingos”.
Other Bingo websites: The “
Wizard
of Odds” also has bingo statistics information - especially
the gambling aspects of Bingo as well as a lot of good stuff on
gambling in general. The probabilities given here match those in the
“Wizard’s” tables. (It's reassuring to have two
independent calculations come up with the same results.)
Return to Durango Bill's Home page
Web page generated via
KompoZer