Probability一問

At a shooting galley, Alan takes shots at a moving target. His performance is affected by the weather. It is given that the probabilities of having a fair, cloudy and rainy days are 0.6, 0.3 and 0.1 respectively. When the days are fair, cloudy and rainy, the probabilities that Alan can hit the target are 0.8, 0.7 and 0.4 respectively. Suppose all shots are independent.

(a)it is given that he fires two shots with both hitting the target. Find the probability that it is a cloudy
Let I be the information:fires two shots with both hitting the target
P(C|I)
=P(I|C)P(C)/[P(I|F)P(F)+P(I|C)P(C)+P(I|R)P(R)]
=(0.7^2)(0.3)/[(0.8^2)(0.6)+(0.7^2)(0.3)+(0.4^2)(0.1)]
=0.2687
(b)
Tthe probability that he hits the target in all shots, given that he hits the target in at least one of the shots.
=P(hits the target in all shots and hits the target in at least one of the shots)/P(hits the target in at least one of the shots.)
=P(hits the target in all shots)/P(hits the target in at least one of the shots.)
=(0.6)^3/(1-0.4^3)
=0.2308
(c)Suppose it is a rainy day and alan wants to assure that the probability of hitting the target at least once greater than 0.9.Find the least number of shots that he must fire.
We wanr
1-0.9^n>0.9
0.1>0.9^n
n>21.8543
So the least number of shots that he must fire is 22