22. A long-term care facility has 500 residents. Among these residents, 200 have dementia, 150 have physical disabilities, and 75 have both dementia and a physical disability. How is the probability of a resident having either dementia or a physical disability but not both calculated?
Answer: C
The probability of a resident having either dementia or a physical disability but not both is calculated as 200/500 + 150/500 - 75/500.
To find the probability of having either dementia or a physical disability but not both, we add the probabilities of each condition and subtract the probability of both conditions being true. This ensures that we do not double count the residents who have both dementia and a physical disability.
A) 1 - (200/500 + 150/500 + 75/500)
This option incorrectly suggests subtracting the total probability of having dementia, physical disabilities, and both from 1. This is not the correct method for calculating the probability of having either condition but not both, as it does not accurately account for the overlap of residents with both conditions.
B) 200/500 + 150/500 + 75/500
This choice sums the probabilities for dementia, physical disabilities, and both conditions without adjusting for overlap. This approach is flawed because it results in double counting the residents who have both conditions, which leads to an inaccurate probability.
C) 200/500 + 150/500 - 75/500
This is the correct calculation because it adds the probabilities of having dementia and physical disabilities while subtracting the probability of having both. By subtracting the overlap, we arrive at the accurate probability of having either condition but not both.
D) 1 - (200/500 + 150/500 - 75/500)
This option misapplies the logic by suggesting that the total of the conditions should be subtracted from 1. It fails to isolate the probability of having either condition without double counting and leads to an incorrect conclusion.
Conclusion
Option C is definitively correct as it properly accounts for the overlap of residents with both dementia and physical disabilities, ensuring an accurate calculation of the probability of having either condition but not both. All other options either miscalculate or misinterpret the probability, making them incorrect.