Question

In: Statistics and Probability

Q2. The patient with a tumor in his brain has decided to have an operation in...

Q2. The patient with a tumor in his brain has decided to have an operation in a private hospital. The hospital has two professors, one professor and the other an associate professor. The patient received data from patients who had surgery on doctors in the past and saw some statistics. There are three alternatives awaiting him after the surgery, such as recovery, temporary partial paralysis and death. 8% of the patients that the professor had previously operated on healed, and 12% suffered temporary partial paralysis. Similarly, 75% of the patients of the associate professor recovered and 21% were subjected to temporary partial paralysis. Until now, 63% of the patients who came to the hospital have operated by professors.
a. What is the probability of death of the patient after surgery?
b. What is the probability that the patient will have surgery by the professor and die?
c. If the patient is known to have temporary partial paralysis, what is the probability that the associate professor will be performed his surgery?

Solutions

Expert Solution

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.


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