You have a torn tendon and are facing arthroscopic surgery to fix it. The surgeon explains the risks of the surgery. Infection occurs in 2% of all cases, and the repair fails in 11% of the cases. 0.5% of the time the repair fails and infection occurs. What is the probability that the operation is successful and infection-free

The probability that  the operation is successful and infection-free is 0.9831

Step-by-step explanation:

Let the infections be denoted by I and the fails be denoted by F then

P(I)= 2% = 0.02

P (F)= 11% = 0.11

P(I∩F)= 0.5 % = 0.005

First we find the probability of infection or fail by using the general rule of addition

P (infection or repair) =  P(I) +P (F) -  P(I∩F)

        = 0.02 + 0.11 -0.005

         = 0.125

Now using the complement method to find the no infection or repair successful

P(no infection or repair successful)= 1- P (infection or repair)

           = 1- 0.125= 0.875

P( Successful) = 1- P (F)= 1- 0.11= 0.89

The probability that  the operation is successful and infection-free is given by  the conditional probability

P( Successful given that no infection) = P(no infection or repair successful)/P(successful)

P( Successful given that no infection) = 0.875/0.89= 0.9831