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.
I see the other answers about the same question, but I still don't understand some of it

Answer 1

To find P(placebo and improvement), we need to use the information given in the problem and apply the formula for conditional probability:

P(A and B) = P(A) x P(B|A)

where P(A) is the probability of event A, and P(B|A) is the conditional probability of event B given that event A has occurred.

In this case, we want to find the probability of two events occurring together: receiving the placebo and reporting an improvement. Let's use a tree diagram to organize the information:

```
A (80%)
/ \
Imp No imp
/ \
B No B
```

From the diagram, we can see that:

- P(placebo) = 1 - P(medication A) = 1 - 0.8 = 0.2
- P(improvement | medication A) = 0.76
- P(no improvement | placebo) = 0.62

Now we can use the formula to find P(placebo and improvement):

P(placebo and improvement) = P(placebo) x P(improvement | placebo)

P(placebo and improvement) = 0.2 x (1 - P(no improvement | placebo))

P(placebo and improvement) = 0.2 x (1 - 0.62)

P(placebo and improvement) = 0.2 x 0.38

P(placebo and improvement) = 0.076

Therefore, the probability of receiving the placebo and reporting an improvement is 0.076, or approximately 0.08, which means that about 8% of the participants received the placebo and reported an improvement.