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.
To find P(placebo and improvement), we need to calculate the probability of receiving a placebo and reporting an improvement.
Let's denote the following probabilities:
P(placebo) = 1 - 0.80 = 0.20
P(improvement | medication A) = 0.76
P(no improvement | placebo) = 1 - 0.62 = 0.38
Then, we can calculate:
P(placebo and improvement) = P(placebo) * P(no improvement | placebo)
P(placebo and improvement) = 0.20 * 0.38
P(placebo and improvement) = 0.076
Therefore, the probability of receiving a placebo and reporting an improvement in the study is 0.076, or 7.6%.