Solve this question step by step. Only in equations, no wording.
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.
Let A be the event of receiving medication A and B be the event of reporting an improvement.
P(A) = 0.80
P(B|A) = 0.76
P(Bᶜ|¬A) = 0.62
P(placebo and improvement) = P(¬A ∩ Bᶜ) = P(¬A) * P(Bᶜ|¬A)
P(placebo and improvement) = (1 - P(A)) * P(Bᶜ|¬A)
P(placebo and improvement) = (1 - 0.80) * 0.62
P(placebo and improvement) = 0.20 * 0.62
P(placebo and improvement) = 0.124
Therefore, P(placebo and improvement) = 0.124
Try again but represent
80% by the letter R
76% by the letter A for medication A
62% bye the letter Y
Let R be the event of receiving medication A and A be the event of reporting an improvement and Y be the event of no improvement.
P(R) = 0.80
P(A|R) = 0.76
P(Y|¬R) = 0.62
P(placebo and improvement) = P(¬R ∩ Y) = P(¬R) * P(Y|¬R)
P(placebo and improvement) = (1 - P(R)) * P(Y|¬R)
P(placebo and improvement) = (1 - 0.80) * 0.62
P(placebo and improvement) = 0.20 * 0.62
P(placebo and improvement) = 0.124
Therefore, P(placebo and improvement) = 0.124
Why the fancy symbols like ∩ and ¬?
Let R be the event of receiving medication A, A be the event of reporting an improvement, and Y be the event of no improvement.
P(R) = 0.80
P(A|R) = 0.76
P(Y|¬R) = 0.62
P(placebo and improvement) = P(¬R ∩ Y) = P(¬R) * P(Y|¬R)
P(placebo and improvement) = (1 - P(R)) * P(Y|¬R)
P(placebo and improvement) = (1 - 0.80) * 0.62
P(placebo and improvement) = 0.20 * 0.62
P(placebo and improvement) = 0.124
Therefore, P(placebo and improvement) = 0.124