(09.02 mc)
arthur is testing the effectiveness of a new acne medication. there are 100 people with acne in the study. forty patients received the acne medication, and 60 other patients did not receive treatment. fifteen of the patients who received the medication reported clearer skin at the end of the study. twenty of the patients who did not receive medication reported clearer skin at the end of the study. what is the probability that a patient chosen at random from this study took the medication, given that they reported clearer skin?

For these types of problems where you are given too many details, I suggest you organize those by means of illustration. For me, I made a tree diagram to map out how many of those 100 people received and did not receive acne, reported and did not reported clearer skin. The diagram is shown in the attached picture.

Next, you have to know the end result of the problem. That is to find the probability that the patient took medication and reported clearer skin. When you want to find the probability of event 1 'AND' event 2 happening, you have to multiply both probabilities. Note that this is a conditional probability. It is stated that it is already given that they reported with clearer skin. This means that the probability that the patient reported a clearer skin is 1. Now, we only have to find the probability of the patient to be given medication. That would be right side of the diagram: 40/100. Therefore, the total probability is

Total probability = 1 × 40/100 
Total Probability = 2/5 or 40%