Suppose that you have torn a tendon and are facing surgery to repair it. The orthopedic surgeon explains the risks to you. Infection occurs in $3\%$of such operations, the repair fails in $14\%$, and both infection and failure occur together $1\%$of the time. What is the probability that the operation is successful for someone who has an operation that is free from infection?

a. 0.8342

b. 0.8400

c. 0.8600

d. 0.8660

e. 0.9900

The Infection occurs in 3% of surgeries, repair fails in 14%, and both infection and failure occur combined in 1% of cases.

Let A signify the occurrence of infection during procedures, and B denote the failure of the repair.

$P\left(A\right)=3/100,P\left(B\right)=14/100,P(Aa<,B)=1/100$

Consider,

$$\begin{gathered} P\left(A^{\text{'}}a^<B^{\text{'}}\right)=P{\left(A{a}^{\wedge a}B\right)}^{\text{'}}=1-P\left(A{a}^{\wedge }{}^{a}B\right) \\ =1-\left[P\left(A\right)+P\left(B\right)-P\left(A{a}^{\wedge }(\subset B)\right]\right. \\ =1-(3/100+14/100-1/100) \\ =1-16/100 \\ =84/100 \\ =0.84 \end{gathered}$$

Option b. is correct

a. The probability that the operation is successful for someone who has an operation that is free from infection 0.8342 is not the correct answer.

c. The probability that the operation is successful for someone who has an operation that is free from infection 0.8600 is not the correct answer.

d. The probability that the operation is successful for someone who has an operation that is free from infection 0.8660 is not the correct answer.

e. The probability that the operation is successful for someone who has an operation that is free from infection 0.9900 is not the correct answer.