# Expected number of calls for a Bingo win

I have read a few similar questions (see this question and this one), but cannot figure out how to adapt the solutions to fit my question. Unfortunately, my understanding of math is extremely limited, and I'm stumped by the number of factors influencing a result here.

I am creating modified BINGO game where players randomly write the numbers 1-42 (instead of 72, as it is normally played) on a 5x5 board (with a free space in the middle). Another change is that they would not write the numbers in order (e.g. the first row L-->R could be 1, 39, 12, 20, 17), and there are no limitations to which numbers can be placed in which rows.Therefore, any square could be any number--they just cannot write the same number twice on the board.

Like the regular (American) version, one letter and one number are drawn for each "call". "BINGO" can be 5 in a row horizontally, vertically, or diagonally.

If 10 people are playing (i.e. 10 boards), what would be the average number of draws it would take to get the first BINGO?

What would be the average number of draws if 20 people were playing?

• Calculating this with a pen-and-paper approach seems entirely too tedious and you would be better off running computer simulations. – JMoravitz Jul 15 at 18:23
• In a call, you random pick a cell and a number to go in the cell, is that right? – saulspatz Jul 15 at 18:24
• What do the letters have to do with anything if the numbers can be anywhere on the grid? In standard Bingo (which goes to $75$) the letters are just to tell you what column the number will be in. If, as in your example, $12$ is in the third column does the player get to mark it if the call is $B12$? – Ross Millikan Jul 15 at 18:24
• As an aside, "game-theory" and related tags are about the mathematics of decision making, not about games like bingo. – JMoravitz Jul 15 at 18:24
• @RossMillikan what made you think that an example call would contain a letter like "B12." It sounds like any such call would be only a number. – JMoravitz Jul 15 at 18:30