Final answer:
The probability of being randomly assigned either a general practitioner or a doctor under the age of 40 is 15/23. This is calculated using the principle of inclusion-exclusion in probability theory by adding the probabilities of each individual event and then subtracting the probability of their intersection.
Step-by-step explanation:
The student asks about the probability of being randomly assigned either a general practitioner or a doctor under the age of 40 at a small hospital. To find this probability, we need to incorporate the principle of inclusion-exclusion from probability theory, which states that for any two events A and B, the probability of A or B is given by P(A) + P(B) - P(A and B).
Let's define the following:
- A = Event that a doctor is a general practitioner
- B = Event that a doctor is under the age of 40
- A and B = Event that a doctor is both a general practitioner and under the age of 40
From the information given:
- P(A) = 5/23 (since there are 5 general practitioners out of 23 doctors)
- P(B) = 13/23 (since there are 13 doctors under the age of 40 out of 23)
- P(A and B) = 3/23 (since 3 doctors are both general practitioners and under the age of 40)
Therefore, the probability of being assigned either a general practitioner or a doctor under the age of 40 is:
P(A or B) = P(A) + P(B) - P(A and B) = (5/23) + (13/23) - (3/23) = 15/23
The answer is 15/23, which can be converted to a decimal if needed, but typically probabilities are left as fractions in exact form.