Bob has two coins,AandB, in front of him. The probability of Heads at each toss isp=0.5for coinAandq=0.9for coinB.

Bob chooses one of the two coins at random (both choices are equally likely).

He then continues with5tosses of the chosen coin; these tosses are conditionally independent given the choice of the coin.

Let:

Hi:the event that Bob'sith coin toss resulted in Heads;

N:the number of Heads in Bob's coin tosses.

Fori{0,1,...,5},pN(i), the pmf ofN, is of the form: 1/2*Binomial[5,a] * b^5 + c * Binomial[5,d] * q^e * (1q)^f.

1. Find the coefficientsa,b,...,f
2. Find E[N]
3. Find var(N | A, H1, H2)
4. Are the events H1 and {N = 5}, independent?
5. Given the 3rd toss resulted in Heads, what is the prob. that coin A was chosen?