Bob has two coins, A and B, in front of him. The probability of Heads at each toss is p=0.5 for coin A and q=0.9 for coin B.
Bob chooses one of the two coins at random (both choices are equally likely).
He then continues with 5 tosses of the chosen coin; these tosses are conditionally independent given the choice of the coin.
Let:
Hi: the event that Bob's ith coin toss resulted in Heads;
N: the number of Heads in Bob's coin tosses.
2. Find E[N] = ?
3. Find the conditional variance of N, in a conditional model where we condition on having chosen coin A and the first two tosses resulting in Heads.
Var(N∣A,H1,H2)= ?
4. Are the events H1 and {N=5} independent? (Yes/No)
5. Given that the 3rd toss resulted in Heads, what is the probability that coin A was chosen?
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