Answer: 58 %
Step-by-step 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,
[tex]P(C)=\frac{n(C)}{n(S)}=\frac{52}{100}=0.52[/tex]
And, the probability of cleared skin of that people who took the medicines,
[tex]P(M\cap C)=\frac{n(M\cap C)}{n(S)}=\frac{30}{100}=0.3[/tex]
Thus, the probability that a patient chosen at random from this study took the medication, given that they reported clearer skin,
[tex]P(\frac{M}{C})=\frac{P(M\cap C)}{n(C)}=\frac{0.3}{0.52}=0.57692307692\approx 0.58 = 58\%[/tex]
Answer: 0.58
Step-by-step 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,
[tex]P(A)=\dfrac{55}{100}=0.55\ \ ,P(B)=\dfrac{45}{100}=0.45[/tex]
[tex]P(C|A)=\dfrac{30}{55}\ \ , P(C|B)=\dfrac{22}{45}[/tex]
Using Bayes theorem, The probability that a patient chosen at random from this study took the medication, given that they reported clearer skin:
[tex]P(A|C)=\dfrac{P(A)\cdot P(C|A)}{P(A)\cdot P(C|A)+P(B)\cdot P(C|B)}\\\\=\dfrac{0.55\cdot\dfrac{30}{55}}{0.55\cdot\dfrac{30}{55}+0.45\cdot\dfrac{22}{45}}=0.576923076923\approx0.58[/tex]
Hence, the required probability : 0.58
