Example Probability and Statistics Questions

  1. Democratic safe
  2. Vacant room
  3. Flip 100 coins
  4. Tennis game
  5. Bankrupt
  1. Combinatorial Analysis
  2. Conditional Probability
  3. Expected Value
  4. General
  5. Markov Chain Probability

Democratic safe

There are 11 friends that found a treasure in the woods. They keep it in a safe. To protect the safe, they decided that only a majority of them together can open the safe. Therefore, they ask a locksmith to put a certain number of locks on the safe. Each lock can have multiple keys, but each key only opens one lock. To access the safe, every lock needs to be opened. the locksmith can give more than one key to each friend. What is the smallest number of locks needed to achieve their democratic goal? And how many keys are needed per person?

Hint

Every subgroup of six friends should have the same key to the same lock that the other five friends do not have.

Answer

When we randomly select five people, there must be a lock that none of them has the key to. Any of the other six friends needs a key to this lock. So, the minimum number of locks needed is \binom{11}{6}=462. Each lock has six keys, which are given to a unique six-member subgroup. The total of 462*6 keys is divided over the 11 friends. So, each person must have 462*6 / 11 = 252 keys.