Chances Bingo

(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 BingoStatistics 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 Bingo4Corners and Letter “X” (both diagonals) link shows the single board probability of getting these patterns after “N” numbers have been called.
The BingoPicture Frame (all 4 edges) and Letter “Y” link shows the single board probability of getting these patterns after “N” numbers have been called.
The BingoProbabilties for a Complete Cover link shows the probabilities for covering all squares on a Bingo board.
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.
Probabilitiesfor 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.)
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