You have torn a tendon and are facing surgery to repair it. The surgeon explains the risks to you: the probability of infection in such operations is 0.05, the probability the repair fails is 0.16, and the probability of both infection and failure is 0.01. What is the probability that the operation succeeds and is free from infection?

Answer: Hello there!

the probability of infection is 0.05 and the probability of failing is 0.16, here you also got the probability of an infection and a failure that is 0.01, which should be equal to the product between the probabilities of infection and of a failure (because both things are happening at the same time), this is 0.05*0.16 = 0.008 (wich is 0.01 if we round it up, wich makes sense!)

then the only probabilities that we need to care about are the one for infection and the one for failure.

the probability in where neither of these things happens (this means that succeeds and is free from infection) is the complement of both of them:

P = 1 - probability of failure - probability of infection = 1 - 0.16 - 0.05 = 0.79

then the probability that you are looking for is P = 0.79

notice that here we are discarding the probability that both things happen, because there is alredy included in each individual probability