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suppose that 5 out of the 23 doctors in a small hospital are general practitioners, 13 out of the 23 are under the age of 40, and 3 are both general practitioners and under the age of 40. what is the probability that you are randomly assigned a general practitioner or a doctor under the age of 40? enter a fraction or round your answer to 4 decimal places, if necessary.

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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.

User Niklas Raab
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