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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 the placebo and reporting an improvement.
Let's denote:
P(placebo) = probability of receiving the placebo = 1 - 0.8 = 0.2
P(improvement|medication A) = probability of improvement given receiving medication A = 0.76
P(no improvement|placebo) = probability of no improvement given receiving the placebo = 0.62
Now, we can calculate P(placebo and improvement) using the formula:
P(placebo and improvement) = P(placebo) * P(no improvement|placebo)
P(placebo and improvement) = 0.2 * 0.62
P(placebo and improvement) = 0.124
Therefore, the probability of receiving the placebo and reporting an improvement is 0.124, or 12.4%.