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?

• 0.60
• 0.63
• 0.66
• 0.69
• 0.72

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.