The kings coinmaster boxes his coins 500 to a box and puts 1

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 1 coin chosen at random out of each of 500 boxes. What is the probability that he finds at least one fake? What is it if the king tests 2 coins from each of 250 boxes?

Solution

PROBABILITY OF FINDING A FAKE COIN = 1/500

PROBABILITY KING FINDS ATLEAST 1 FAKE COIN = 1 - PROBABILITY HE DOES NOT FIND ANY FAKE COIN AMONG 500 BOX

= 1 - (499/500)^500 = 0.634

PROBABILITY KING FINDS ATLEAST 1 FAKE COIN IF CHOOSE 2 = 1 - PROBABILITY HE DOES NOT FIND ANY FAKE COIN

= 1 -(498/500)^250 = 0.632