When coin 1 is flipped, it lands on heads with probability .4; when coin 2 is flipped, it lands on heads with probability .7. One of these coins is randomly chosen and flipped 10 times.
(a) What is the probability that the coin lands on heads on exactly 7 of the 10 flips?
(b) Given that the first of these 10 flips lands heads, what is the conditional probability that exactly 7 of the 10 flips land on heads?
I know how to to do by finding the probability of coin 1 being chosen then finding the probability it gets exactly 7 heads by using a bernoulli sum and adding that probability to the probability of coin 2 being chosen and doing another bernoulli sum for that coin, but is there a better or more efficient way of calculating the probability of part a instead of doing it the long way like I did?
and part b does not make sense to me because if it's given that the first 10 flips are heads how can you find the probability that exactly 7 of the 10 landed on heads if it's given that they all landed on heads?