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.
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
Received placebo (1-.80) = .20
Showed improvement (1-.62) = .38
To find P(placebo and improvement), we need to find the probability of two events happening together: receiving a placebo and reporting an improvement.
Let's solve this step-by-step:
Step 1: Find the probability of receiving a placebo.
Given that 80% received medication A, the probability of receiving a placebo is 100% - 80% = 20%.
Step 2: Find the probability of reporting an improvement given the placebo.
Given that 62% reported no improvement, the probability of reporting an improvement is 100% - 62% = 38%.
Step 3: Multiply the probabilities from step 1 and step 2.
P(placebo and improvement) = 20% × 38% = 0.20 × 0.38 = 0.076 or 7.6%.
Therefore, the probability of receiving a placebo and reporting an improvement is 7.6%.
To find the probability P(placebo and improvement), we need to calculate the probability that a participant received the placebo and reported an improvement. We can break it down into two parts:
1. Finding the probability of receiving a placebo:
Given that 80% of the participants received medication A, we can calculate the probability of receiving a placebo as:
P(placebo) = 1 - P(medication A)
= 1 - 0.80
= 0.20
So, the probability of receiving a placebo is 0.20 or 20%.
2. Finding the probability of improvement given the placebo:
Given that 62% of those who received the placebo reported no improvement, we can calculate the probability of improvement given the placebo as:
P(improvement | placebo) = 1 - P(no improvement | placebo)
= 1 - 0.62
= 0.38
So, the probability of improvement given the placebo is 0.38 or 38%.
Now, to find the probability of both receiving a placebo and reporting improvement, we can multiply the probabilities:
P(placebo and improvement) = P(placebo) * P(improvement | placebo)
= 0.20 * 0.38
= 0.076
So, the probability of receiving a placebo and reporting improvement is 0.076 or 7.6%.