Discrete Probability spaces

due by Sep 3, 2018

     A fair coin is tossed until two consecutive tosses have the same result.

1. Describe the sample space $\Omega$.
2. Give a formula for the function $p:\Omega \to \mathbb{R}$ giving the probabilities of outcomes.
3. Show that $(\Omega, p)$ is a discrete probability space, in particular show that $\sum\limits_{\omega\in\Omega}p(\omega) = 1.$
4. What is the probability that the number of tosses (until two consecutive tosses have the same result) is at most $6$?
5. What is the probability that the number of tosses is even?

     A nine digit number is formed by arranging the digits $1$ to $9$ in a random order, with equal probability for each permutation. Let $A_i$ be the event that the $i$th digit is $i$ (we say the $i$th digit is in the correct place) and $A$ the event that some digit is in the correct place.

6. By relating $A$ to the events $A_i$, obtain an upper bound on $P(A)$. Is this a useful bound?
7. Compute $P(A_i\cap A_j)$ and obtain a lower bound on $P(A)$ using this. Why do you think (intuitively) that this bound is not very good, i.e., close to the upper bound?