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 can break it down into two parts:
1. Probability of receiving placebo: 20% (since 80% received medication A, the remaining 20% received the placebo)
2. Probability of improvement given placebo: 38% (since 62% reported no improvement, the remaining 38% reported improvement)
Therefore, the probability of receiving the placebo and reporting an improvement is:
P(placebo and improvement) = P(placebo) * P(improvement|placebo)
P(placebo and improvement) = 0.20 * 0.38
P(placebo and improvement) = 0.076
So, the probability of receiving the placebo and reporting an improvement is 0.076, or 7.6%.