Problem:
Forty two cards are labeled with the natural numbers through and randomly shuffled into a stack. One by one, cards are taken off of the top of the stack until a card labeled with a prime number is removed. How many cards are removed on average?
Solution:
Answer: . Note that there are prime numbers amongst the cards. We may view these as separating the remaining cards into groups of nonprimes - those appearing before the first prime, between the first and second, etc. Each of these groups is equally likely to appear first, so nonprimes are removed on average. We are done since exactly one prime is always drawn.