The king's coinmaster boxes his coins 500 to a box and puts 1 counterfeit coin in each box. The king is suspicious, but, instead of testing all the coins in 1 box, he tests 2 coins at random from each of 250 boxes. What is the probability that he finds at least one fake?

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

Solution: B

There are 500 coins in each box with one fake, so the probability that he finds at least one fake when he randomly selects 2 coins from a single box equals 1-(499/500)*(498/499) = 0.004.

Then the number of fakes that he finds after testing 250 boxes is a binomial distribution with parameters [math]n=250, p=0.004 [/math] and therefore the probability that he finds at least one fake is 1 minus the probability of not finding any fakes: 1-(1-0.004)²⁵⁰ = 0.6329.