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BINGO! (Posted on 2013-03-07) Difficulty: 3 of 5
The game BINGO! (US and Canada version)is played with a card on which is drawn a 5x5 grid filled with 25 random, non-repeating integers from the range of 1 to 75 inclusive (1 number per grid location). Further constraints are that the left-most column can only contain values from 1 to 15 inclusive, the next left-most column only 16 to 30 inclusive, etc., ending with the 5th (right-most) column only containing 5 random values from the range 61 to 75 inclusive. To play, numbers are "called", one at a time and randomly, with the winner being the first to have all values in any single row, column or diagonal on his/her card "called" (thus, a BINGO!).

Without referring to the many solutions available on the web (BINGO! is very popular):

1) What is the expected value of the number of numbers that must be "called" to reach a BINGO! on a single card?

2) What if you are allowed to play 5 cards (presumably all different)?

Also, usually the center square of the grid is "free", i.e. it is assumed to be called already at the beginning of the game.

3) What are the resulting values for Questions 1) and 2) in this case?

Extra hard bonus:

There are many many variations of the game that allow changes to the pattern of numbers/grid spaces that must be "called" to reach a BINGO! A "+" and an "X" are two of these. Both require 8 numbers to be called, assuming there is a free space in the middle. What are the expected values of the number of numbers "called" for each of these?

The author admits that computer solutions are very viable to solve this problem, but BINGO! existed long before computers. Any analytical attempts/solutions get bonus points!

No Solution Yet Submitted by Kenny M    
Rating: 5.0000 (1 votes)

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Charlie's analysisKenny M2013-03-08 22:16:26
Charlie's analysisKenny M2013-03-08 22:16:25
Solutiondifferent program -- agrees with previous resultsCharlie2013-03-08 12:03:14
Some ThoughtsSimulation resultsCharlie2013-03-07 22:42:19
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