Question

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?

Answer 1

I have a tree diagram started

Answer 2

The first step is to calculate the probability of failure, given that there is infection.
\[\large P(F \cap I)=0.005\]
\[\large P(F|I)=\frac{P(F \cap I)}{P(I)}=\frac{0.005}{0.02}=0.25\]
Then the probability of success, given infection occurs is
\[\large P(S|I)=1-P(F|I)=1-0.25=0.75\]
Are you able to follow so far?

Answer 3

Yes, but does it not give you the probability of failure and infection? I thought that's what the '0.5% of the time the repair fails and infection occurs,' meant....

Answer 4

The probability of success and infection is given by
\[\large P(S \cap I)=P(I) \times P(S|I)=0.02\times0.75=0.015\]
We are given that the repair fails in 11% of all cases, therefore the overall probability of success is
1 - 0.11 = 0.89
We have found that the probability of success and infection is 0.015. Therefore the probability of success and no infection is
0.89 - 0.015 = 0.875
A corrected drawing follows.

Answer 5

That makes sense..

Answer 6

That's good :)