The probability of a patient having insurance coverage from Medicare is 45%, and the probability of a patient having insurance coverage from Medicaid is 25%. The probability of a patient having insurance coverage from both Medicare and Medicaid is 15%. What is the probability that the patient has insurance coverage from either Medicare or Medicaid?

Rationale

The probability that the patient has insurance coverage from either Medicare or Medicaid is 55%.

To find the probability of a patient being covered by either Medicare or Medicaid, we use the formula for the union of two events: P(Medicare ∪ Medicaid) = P(Medicare) + P(Medicaid) - P(Medicare ∩ Medicaid). Plugging in the numbers gives us 0.45 + 0.25 - 0.15 = 0.55, or 55%.

A (85%) YOUR ANSWER

This option incorrectly suggests that the combined probability is significantly higher than the actual calculation. Adding the individual probabilities without accounting for the overlap (those covered by both) results in an overestimation, which is why this value is not correct.

B (55%) CORRECT

This is the correct answer as it accurately reflects the probability of a patient having coverage from either Medicare or Medicaid. The calculation correctly considers the overlap of patients covered by both programs, ensuring no double-counting occurs.

C (40%) YOUR ANSWER

Choosing this probability indicates a misunderstanding of the union of probabilities. It appears to subtract the intersection from each individual event incorrectly, which leads to an underestimation of the coverage probability.

D (70%) YOUR ANSWER

This option also overestimates the probability by incorrectly accounting for the overlap. It may arise from an erroneous addition of the individual probabilities without properly subtracting the intersection, leading to a misleading total.

Conclusion

The probability of a patient having insurance coverage from either Medicare or Medicaid is correctly calculated to be 55%, ensuring that the overlap of the two groups is considered. This principle of probability helps maintain accurate assessments in healthcare coverage statistics, preventing double-counting and ensuring clearer insights into patient insurance status.

The probability that the patient has insurance coverage from either Medicare or Medicaid is 55%.

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