Question

4.10 13.31 tendon surgery. you have torn a tendon and are facing surgery to repair it. the 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 in 1%. what percentage of these operations succeed and are free from infection? follow the four - step process in your answer.

Explanation:

Step1: Use the principle of inclusion - exclusion

Let \(A\) be the event of infection and \(B\) be the event of repair failure. We know \(P(A)=0.03\), \(P(B) = 0.14\), and \(P(A\cap B)=0.01\). The probability of either infection or repair failure or both is given by \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\).

$$P(A\cup B)=0.03 + 0.14-0.01$$

$$P(A\cup B)=0.16$$

Step2: Find the probability of success and no - infection

The probability of an operation being successful and free from infection is the complement of the event \(A\cup B\). Let \(P(\text{success and no infection})\) be \(P(\overline{A\cup B})\). Since \(P(\overline{E})=1 - P(E)\) for any event \(E\), we have \(P(\overline{A\cup B})=1 - P(A\cup B)\)

$$P(\overline{A\cup B})=1-0.16$$

$$P(\overline{A\cup B}) = 0.84$$

Answer:

\(84\%\)