We have two biased coins. The first one yields heads with probability 0.1 and the second one yields heads with probability 0.9. We choose one of the two coins randomly (with probability 0.5 each; we can't tell them apart by sight). That coin is then tossed 10 times in a row. Let N denote the number of heads we see in these 10 tosses. Find the conditional expectation of N given that there were exactly 2 heads in the first 3 tosses.
So, I understand how you would calculate the conditional expectation if you were given ONE biased coin and asked the expected number of times it lands on heads, but how do you account for randomly picking between two coins? And how do you account for exactly "2 heads in the first 3 tosses"?