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

P(placebo and improvement) = 0.076 or 7.6%.

Step-by-step explanation:

We are given that Participants in a study of a new medication received either medication A or a placebo.

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 Probability that participants received medication A = P(M) = 0.80

Probability that participants received placebo = P(P) = 1 - P(M) = 1 - 0.80 = 0.20

[Because there are only two cases either medication A or a placebo]

Let I = event that there is an improvement

Also, Probability that participants reported improvement given that they had received medication A  =  P(I / M) = 0.76

Probability that participants reported no improvement given that they had received placebo  =  P(I' / P) = 0.62

So, Probability that participants reported improvement given that they had received placebo  =  P(I / P) = 1 - P(I' / P) = 1 - 0.62 = 0.38

Now, Probability of (placebo and improvement) = Probability that participants received placebo  $\times$  Probability that participants reported improvement given that they had received placebo

                          =  P(P)  $\times$  P(I / P)  

                          =  0.20 $\times$ 0.38  =  0.076  or  7.6%

Therefore, the required probability is 7.6%.