19. A town has 78 primary care practices: 33 employ a physician's assistant, 23 employ a nurse practitioner, 18 employ both a physician's assistant and nurse practitioner. How is the probability of a primary care practice employing either a physician's assistant or a nurse practitioner calculated?
Answer: D
The probability of a primary care practice employing either a physician's assistant or a nurse practitioner is calculated as 33/78 + 23/78 - 18/78.
To find the probability of a primary care practice employing either a physician's assistant or a nurse practitioner, we use the formula for the union of two sets. This involves adding the probabilities of each individual event and subtracting the probability of both events occurring together to avoid double counting.
A) 1 - (33/78 + 23/78 - 18/78)
This option incorrectly suggests subtracting the calculated probability from 1. The question requires the direct calculation of the probability of employing either a physician’s assistant or a nurse practitioner rather than its complement.
B) 1 - (33/78 + 23/78 + 18/78)
Similar to Option A, this option incorrectly indicates that we should subtract the total of all probabilities from 1. The formula does not account for the overlap correctly and thus does not provide the probability of employing either type of practitioner.
C) 33/78 + 23/78 + 18/78
This option fails to account for the overlap where both a physician's assistant and a nurse practitioner are employed in a practice. By simply adding the three probabilities, it incorrectly inflates the total probability, leading to an inaccurate result.
D) 33/78 + 23/78 - 18/78
This is the correct calculation as it properly applies the principle of inclusion-exclusion. By adding the probabilities of employing a physician's assistant and a nurse practitioner and then subtracting the overlap (practices that employ both), it accurately calculates the probability of employing either type of practitioner.
Conclusion
Option D is definitively correct as it correctly applies the inclusion-exclusion principle to calculate the probability of a primary care practice employing either a physician’s assistant or a nurse practitioner. All other options fail to correctly account for the overlap or use incorrect operations, leading to inaccurate results.