Participants in a study of a new medication received either medication 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 first find the probability of receiving a placebo and then the probability of reporting an improvement given that a placebo was received.
P(placebo and improvement) = P(placebo) * P(improvement | placebo)
First, let's find P(placebo):
P(placebo) = 1 - P(medication A) = 1 - 0.80 = 0.20
Next, let's find P(improvement | placebo):
P(no improvement | placebo) = 0.62
Therefore, P(improvement | placebo) = 1 - P(no improvement | placebo) = 1 - 0.62 = 0.38
Now we can find P(placebo and improvement):
P(placebo and improvement) = P(placebo) * P(improvement | placebo) = 0.20 * 0.38 = 0.076
Therefore, the probability of someone receiving a placebo and reporting an improvement is 0.076 or 7.6%.