Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
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.
STEP BY STEP WORK
Step 1: Calculate the probability of receiving a placebo.
Probability(Placebo) = 1 - Probability(Medication A)
Probability(Placebo) = 1 - 0.80
Probability(Placebo) = 0.20
Step 2: Calculate the probability of improvement for those who received medication A.
Probability(Improvement | Medication A) = 0.76
Step 3: Calculate the probability of no improvement for those who received the placebo.
Probability(No Improvement | Placebo) = 0.62
Step 4: Calculate the probability of both receiving the placebo and reporting an improvement using the formula for the intersection of two events.
P(Placebo and Improvement) = P(Placebo) * P(Improvement | Medication A)
P(Placebo and Improvement) = 0.20 * 0.76
P(Placebo and Improvement) = 0.152
Therefore, the probability of a participant receiving a placebo and reporting an improvement is 0.152 or 15.2%.