In: Statistics and Probability
The hospital has two professors and these are a professor and an associate professor. The outcomes of surgery are recovery, temporary partial paralysis, and death.
The probability of recovery when operated by professor is P(R|P) = 0.08. The probability of temporary partial paralysis, when operated by the professor, is P(T|P) = 0.12. Hence, the probability of death when operated by professor becomes P(D|P) = 1 - P(R|P) - P(T|P) = 1 - 0.08 - 0.12 = 0.80.
The probability of recovery, when operated by an assistant professor, is P(R|A) = 0.75. The probability of temporary partial paralysis, when operated by the assistant professor, is P(T|A) = 0.21. Hence, the probability of death when operated by assistant professor becomes P(D|A) = 1 - P(R|A) - P(T|A) = 1 - 0.75 - 0.21 = 0.04.
a) We are to determine the probability of a patient dying after a surgery. This can happen after undergoing surgery by either the professor or assistant professor. Hence, the probability of death will be:
P(D) = P(D|P) + P(D|A) = 0.80 + 0.04 = 0.84.
b) The probability that the patient will have surgery by professor and die will be P(D|P) = 0.8
c) We are lastly to determine the probability that given the patient has temporary paralysis, what is the probability that the patient underwent a surgery by the associate professor. This is the probability P(A|T).
This can be calculated using Bayes' Theorem.
Hence, we have:
Now, the P(T|A') = P(T|P) = 0.12
And similarly, P(A') = P(P). Notice that we are not given the probabilities P(A) and P(P) and hence this problem cannot be solved.