Question

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.

1. For i∈{0,1,…,5}, pN(i), the pmf of N, is of the form
1/2(5 C a)b^5+c(5 C d) q^e(1−q)^f.

Find the coefficients a,b,…,f. Your answer can be either a number or an expression involving i.
a = ?
b = ?
c = ?
d = ?
e = ?
f = ?