Answer:
0.4007
Step-by-step explanation:
Let's define the following events:
A: method A is used
B: method B is used
NR: the eye has not recovered in a month
R: the eye is recovered in a month
The probability that the eye has not recovered in a month is 0.002 if method A is used, i.e., P(NR|A) = 0.002, so P(R|A) = 0.998.
When method B is used, the probability that the eye has not recovered in a month is 0.005, i.e., P(NR|B) = 0.005, so P(R|B) = 0.995.
40% of eye surgeries are done with method A, i.e., P(A) = 0.4
60% of eye surgeries are done with method B, i.e., P(B) = 0.6
If an eye is recovered in a month after surgery is done in the hospital, what is the probability that method A was performed? We are looking for P(A|R), then, by Bayes' Formula
P(A|R) = P(R|A)P(A)/(P(R|A)P(A) + P(R|B)P(B)) = 0.998*0.4/(0.998*0.4 + 0.995*0.6) = 0.4007
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