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Question

Participants in a study of a new medication received either medication A or a placebo. Find P(placebo and improvement). You may find it helpful to make a tree diagram of the problem on a separate piece of paper.
Of all those who participated in the study, 80% received medication A.

Of those who received medication A, 76% reported an improvement.

Of those who received the placebo, 62% reported no improvement.

Answer 1

Let P(A) = 0.80
Let P(improvement | A) = 0.76
Let P(no improvement | placebo) = 0.62

P(placebo and improvement) = P(placebo) * P(improvement | placebo)
P(placebo and improvement) = (1 - P(A)) * (1 - P(no improvement | placebo))
P(placebo and improvement) = (1 - 0.80) * (1 - 0.62)
P(placebo and improvement) = 0.20 * 0.38
P(placebo and improvement) = 0.076

Therefore, P(placebo and improvement) = 0.076