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Six problems related to probability and combinatorics. The problems involve expected number of people picking up their own coat, number of k-partitions of a non-negative integer, distinct 3-peats in flipping coins, distinct pairings in a tennis tournament, probability of seeing an even number of heads in flipping coins, and proving equalities and properties of fibonacci numbers.
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N people that pick up their coats after a performance. Unfortunately, the claim tickets are lost, so each person picks one coat at random. What is the expected number of people who pick up their own coat?
We call a k-parition of a non-negative integer N a sequence of k non-negative integers, a 1 , a 2 ,... , ak− 1 , ak, that sums to N. Given k and N , how many k-partitions are there of N? Give a formula for this and prove it correct.
Suppose 2n players enter a tennis tournament. In the first round, each player plays against one other. This is called a “pairing.” How many distinct pairings are there? Provide a formula and show it is correct.
If you flip n coins in a row, what is the probability you see an even number of heads? Hint: Find a formula for
∑ kiseven
(n k
)
. Prove your result.
