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?

Rationale

33/78 + 23/78 - 18/78

To calculate the probability of a primary care practice employing either a physician's assistant or a nurse practitioner, we use the principle of inclusion-exclusion. This involves adding the probabilities of each group and subtracting the probability of their overlap, which in this case is the practices employing both.

A (1 - (33/78 + 23/78 - 18/78)) YOUR ANSWER

This choice incorrectly applies the complement rule by subtracting the union of the two groups from 1. The probability of employing either a physician's assistant or a nurse practitioner should be calculated directly, not through the complement.

B (1 - (33/78 + 23/78 + 18/78)) YOUR ANSWER

This option also misapplies the complement rule and incorrectly sums all probabilities, including the overlap. It does not reflect the accurate calculation for the probability of employing either type of provider.

C (33/78 + 23/78 + 18/78) YOUR ANSWER

This choice inaccurately adds the probabilities of all groups without accounting for practices that employ both a physician's assistant and a nurse practitioner. This results in double counting those practices, leading to an inflated probability.

D (33/78 + 23/78 - 18/78) CORRECT

This is the correct approach as it accurately accounts for the overlap of practices employing both a physician's assistant and a nurse practitioner. By subtracting the overlap, we avoid double counting and obtain the correct probability for practices employing either provider.

Conclusion

The calculation of the probability of a primary care practice employing either a physician's assistant or a nurse practitioner is fundamental in understanding the distribution of healthcare providers. By using the inclusion-exclusion principle, specifically the formula 33/78 + 23/78 - 18/78, we ensure that practices employing both types are counted only once, leading to an accurate probability representation.

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?

33/78 + 23/78 - 18/78

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