# Spring 2018, Math 171

#### Week 1

I flip a coin with probability $$p$$ of heads until the first head occurs. Let $$N$$ be the number of times I flip this coin. I then flip a coin with probability $$q$$ of heads $$N$$ times. Let $$H$$ be the number of heads which occur with the second coin.

1. (Discussed) Compute $$P(N=n)$$

2. (Discussed) Compute $$P(H=0)$$

• Hint: Law of Total Probability
3. (Discussed) Compute $$P(N=n|H=0)$$

• Hint: Bayes Formula
4. (Discussed) Compute the distribution of $M(N) = \begin{cases}\frac{N}{2}, &\text{if } n \text{ is even} \\ \frac{N+1}{2}, &\text{if } n \text{ is odd} \end{cases}$

5. (Discussed) Show memorylessness: $$P(N=m+n|N>m) = P(N=n)$$

• Hint: Use the definition of Conditional Probability