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 find the probability of someone receiving the placebo and reporting an improvement.
Let:
- P(Medication A) = 0.80
- P(Improvement | Medication A) = 0.76
- P(No Improvement | Placebo) = 0.62
We can start by finding the probability of receiving the placebo, which is the complement of receiving Medication A:
P(Placebo) = 1 - P(Medication A) = 1 - 0.80 = 0.20
Now, we need to find the probability of someone receiving the placebo and reporting an improvement. This can be calculated using the following formula:
P(Placebo and Improvement) = P(Placebo) * P(Improvement | Placebo)
P(Placebo and Improvement) = 0.20 * (1 - P(No Improvement | Placebo)) = 0.20 * (1 - 0.62) = 0.20 * 0.38 = 0.076
Therefore, P(placebo and improvement) is 0.076 or 7.6%.