9/2/2009

Each box of a particular brand of cereal contains one of a set of n different plastic animals. Suppose that the animal in each box is equally likely to be any one of the set of n, independently of what animals are in other boxes. What is the expected number of cereal boxes a collector must buy in order to obtain a complete set of animals?

Mathematical Approach
Based on the sum of geometric random variables. Let Xi denote the number of additional boxes required to get i different animals once (i-1) have been obtained. Thus T = X1 + X2 + … + Xn.

Multiplication Principle
- Application 1 - Rolling Dice
- Application 2 - Tossing a Coin k times
Permutations
Combinations
- Example 1 - In how many ways can I arrange 8 students into 8 chairs?
- Example 2 - In how many ways can I put 8 students into 3 chairs?
- Example 3 - Suppose there are 4 women and 4 men and I am interested in filling the three chairs in the following way: Woman-Man-Woman. In how many ways can I do this?
- Example 4 - How many permutations exist for the letters in the word statistics?
- Example 5 - Consider a bowl which contains 6 ping pong balls numbered 1, 2, 3, 4, 5, and 6. If we sample 3 balls without replacement,
  - how many sequences of outcomes are possible?
  - What is the probability that the first ball is a 2?
  - What is the probability that the first ball is i (for i=1, 2, ..., 6)?
  - What is the probability that the second ball is i (for i = 1, 2, ..., 6)?
- Example 6 - A personnel officer has available 8 candidates, five men and three women, to fill 4 positions. If every combination of candidates is equally likely to be chosen, what is the probability that no women will be hired?
- Example 7 - How many committees are possible from a group of five individuals when the size of the committee is unspecified