Playing Bingo on a recent holiday they gave out a booby prize is you had no numbers marked off your card after ten numbers had been called, before continuing with play.
One day one player still had no numbers marked off after 13 numbers had been called - we know this because someone got a line at that point.
Later in the holiday someone had no numbers marked off after ten numbers had been called on BOTH cards that they were playing.
A little debate started as to which of this occurrences was the most unlikely.
Each card had 24 numbers on it and the numbers went from 1 to 75.
So which is it?
Taking the first case there are 75 - 24, that is 51, numbers not on the card at the start, so the odds of the first number called not being on the card is 51/75. The odds of the second number called not being on the card is 51/74, so the odds of neither number being on the card is (51/75) x (51/74). Continue this process for 13 numbers and the probability is 0.12% or around 1 in 800.
With the second case things are a little complicated by the possibility of one or more numbers being on both cards. Assuming for the moment that not numbers are shared, there are 75 - 48, that is 27, numbers not on the cards at the start, so the odds of the first number called not being on the card is 27/75. The odds of the second number called not being on the card is 27/74, so the odds of neither number being on the card is (27/75) x (27/74). Continue this process for 13 numbers and the probability is 0.0000053% or around 1 in 20 million!
Clearly a win for the second case, so let us assume some numbers are shared - six was the most we saw.
In this situation, there are 75 - 42, that is 33, numbers not on the cards at the start, so the odds of the first number called not being on the card is 33/75. The odds of the second number called not being on the card is 33/74, so the odds of neither number being on the card is (33/75) x (33/74). Continue this process for 13 numbers and the probability is 0.00015% or around 1 in 670,000!
From this we can see that even with a lot of numbers shared (and there must have been a lot of numbers shared) the second case is far, far more unlikely.