### Spies at Emperor Wang Shu’s Court

Saturday, December 31st, 2016One summer evening during the hour of the horse, the Chinese emperor Wang Shu summoned his spymaster Lang Dang to his private suite in the palace. He was worried, as he had received information from the palace cook Chu Shi that his enemy, the warlord Dui Yuan, had planted ten spies among his royal guards.

“Lang Dang, there are ten spies among my hundred royal guards. They all have a dragon tattoo on the sole of their right foot. You are to find at least one before the hour of the rat,” commanded Wang Shu.

“Yes, heavenly ruler, this is already done,” said Lang Dang with a deep bow, his thin grey beard reaching large black sleeves into which he had stuck his hands.

“You will interrogate Dui Yuan’s spies you discover so as to reveal the identities of the others. You should know that the leader of the spy ring has a dragon tattoo on the soles of both feet,” added emperor Wang Shu. “Find him and Mei Ling is yours as a concubine,” he offered.

“Ah so.” Lang Dang retreated bowing deeply and shuffled off eagerly, making some mental calculations.

To avoid arousing any alarm and achieve rapid results, Lang Dang decided to round up ten royal guards at random and check their feet. That should do the trick. He would interrogate them until their leader was revealed.

*What is the probability of Lang Dang finding one or more of Dui Yuan’s spies on selecting ten royal guards for foot inspection?*

*How many royal guards must Lang Dang select to be 90% sure to find a spy?*

*What is the probability of Lang Dang finding the leader of Dui Yuan’s spy ring in a sample of ten royal guards?*