Question

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?
A. 15%
B. 33%
C. 38%
D. 43%

Answer 1

Here is my answer below.

Answer 2

Answer:  D. 43 %

Step-by-step explanation:

Let M is the event of receiving medicine,

M' is the event of not receiving medicine,

C is the event of clear skin.

Then According to the question,

Total size of the sample space, n(S) = 100

Number of patient who get the medicine, n(M) = 40

Number of patient who do not get the medicine, n(M') = 60

Number of patient  who received the medication reported clearer skin at the end of the study, $n(M\capC) = 15$

Therefore, the probability that patient  who received the medication reported clearer skin at the end of the study, $P(M\cap C) = \frac{n(M\cap C)}{n(S)} = \frac{15}{100} = 0.15$

Number of patient who who did not receive the medication reported clearer skin at the end of the study, n(M\capC) = 20.

Thus, the Number of patient who cleared the skin, n(C) = 15 + 20 = 35

And, the probability that the patient cleared their skin,  $P(C) = \frac{n(C)}{n(S)} = \frac{35}{100} = 0.35$

Therefore, the probability that a patient chosen at random from this study took the medication, given that they reported clearer skin,

$P(\frac{M}{C} ) = \frac{P(M\cap C)}{P(C)}$

⇒ $P(\frac{M}{C} ) = \frac{0.15}{0.35}$

⇒  $P(\frac{M}{C} ) = \frac{3}{7}$

⇒   $P(\frac{M}{C} ) =0.42857142857\approx 0.43$

Thus,  $P(\frac{M}{C} ) = 0.43$ or $43\%$