Question

Maryann is testing the effectiveness of a new acne medication. There are 100 people with acne in the study. Fifty-five patients received the acne medication, and 45 other patients did not receive treatment. Thirty of the patients who received the medication reported clearer skin at the end of the study. Twenty-two 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?

Answer 1

Answer: 0.58

Explanation:

Let A = Event that the patients received the acne medication.

B = Event that the patients did not receive the acne medication.

C = Patient reported reported clearer skin.

Now,

`P(A)=(55)/(100)=0.55\ \ ,P(B)=(45)/(100)=0.45`

`P(C|A)=(30)/(55)\ \ , P(C|B)=(22)/(45)`

Using Bayes theorem, The probability that a patient chosen at random from this study took the medication, given that they reported clearer skin:

`P(A|C)=(P(A)\cdot P(C|A))/(P(A)\cdot P(C|A)+P(B)\cdot P(C|B))\\\\=(0.55\cdot(30)/(55))/(0.55\cdot(30)/(55)+0.45\cdot(22)/(45))=0.576923076923\approx0.58`

Hence, the required probability : 0.58

Answer 2

Answer: 58 %

Explanation:

Let M represents the event of taking medicine, M' represents the event of not taking medicine and C represents the event of clearing skin,

Thus, according to the question,

n(M) = 55,

n(M') = 45,

n(M∩C) = 30,

n(M'∩C)= 22,

⇒ n(C) = n(M∩C) + n(M'∩C) = 30 + 22 = 52

Let S shows the total number of people,

⇒ n(S) = 100

Hence, the probability of cleared skin,

`P(C)=(n(C))/(n(S))=(52)/(100)=0.52`

And, the probability of cleared skin of that people who took the medicines,

`P(M\cap C)=(n(M\cap C))/(n(S))=(30)/(100)=0.3`

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

`P((M)/(C))=(P(M\cap C))/(n(C))=(0.3)/(0.52)=0.57692307692\approx 0.58 = 58\%`