A group of psychiatrists has noticed an unusual number of older individuals with depression in the surrounding community. A decision is taken to conduct a cross-sectional study to estimate the prevalence of depression in this specific population. Estimates for the prevalence of depression (cases per 100 individuals) in the community are as follows: Suppose during an afternoon clinic, a psychiatrist sees a 65-year-old woman, a 72-year-old man, and an 84-year-old woman from this community. Assuming no relation between the 3 patients, what is the probability that none of them has depression?

Question

A group of psychiatrists has noticed an unusual number of older individuals with depression in the surrounding community.  A decision is taken to conduct a cross-sectional study to estimate the prevalence of depression in this specific population.  Estimates for the prevalence of depression (cases per 100 individuals) in the community are as follows:  Suppose during an afternoon clinic, a psychiatrist sees a 65-year-old woman, a 72-year-old man, and an 84-year-old woman from this community.  Assuming no relation between the 3 patients, what is the probability that none of them has depression?

Quizplus · Accepted Answer

11ec160b_3716_1ca4_9249_7b940e7b12f8_MD0001_00 The age- and gender-specific prevalence of depression in the community is a good estimator of the probability that a corresponding individual from community has the disease. Because each of the 3 patients is unrelated to the others, each of the diagnoses is an independent event (ie, a patient's diagnosis has no impact on another's). When events are independent of each other, the probability that all the events occur is calculated by multiplying together the probability of each individual event. In this case, to calculate the probability that none of the 3 patients has depression, multiply the probability of each complementary event (ie, the probability of not having depression): (1 − 0.091) × (1 − 0.104) × (1 − 0.135) = 0.909 × 0.896 × 0.865 = 0.705 (Choices A and E) The probability that all 3 patients have depression can be calculated by multiplying the probability of each event, (0.091) × (0.104) × (0.135) = 0.001. The complementary event (ie, the probability that at least 1 patient does not have depression) can be calculated by subtracting the probability that all 3 patients have depression from 1, or 1 − [(0.091) × (0.104) × (0.135)] = 0.999. (Choice B) The probability that only the women have depression can be calculated by multiplying together the probability of each woman having the disorder along with the probability of the man not having the disorder, (0.091) × (0.135) × (1 − 0.104) = 0.011. (Choice C) The probability that at least 1 patient has depression is complementary to the probability of none of the 3 patients having depression. It can be calculated by subtracting the probability that none of the 3 patients has depression from 1, or 1 − [(1 − 0.091) × (1 − 0.104) × (1 − 0.135)] = 0.295. Educational objective: The multiplication law of probability states that the probability of 2 or more independent events occurring together can be calculated by multiplying the individual probabilities of each event. The multiplication law of probability can be extended to complementary events (1 − probability of event) in the same manner.

A Group of Psychiatrists Has Noticed an Unusual Number of Older