Question

Colby is testing the effectiveness of a new allergy medication. There are 100 people with allergies in the study. Sixty-two patients received the allergy medication, and 38 other patients did not receive treatment. Fifty of the patients who received the medication reported no allergies at the end of the study. Twenty of the patients who did not receive medication reported no allergies 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 no allergies?

Answer 1

Simply put, since we are *given* that the person reported no allergy, the group of interest is 50 medicated and 20 non-medicated 

indicated Pr is thus = 50/70 = 5/7, 71% to nearest %

Answer 2

Answer: $\frac{5}{7}$

Step-by-step explanation:

Here, the size of sample space = 100 patient.

Let M shows the event that the patients received the allergy medication,

M' shows the event that the patients do not received the allergy medication,

And, N shows the event that the patient does not has the allergy.

Thus, According to the question,

$P(M)=\frac{62}{100} = 0.62$

$P(M')=\frac{38}{100} = 0.38$

$P(N)=\frac{50+20}{100} = \frac{70}{100}=0.70$ ( Because, Total number of patient who do not have allergy = total patients who received the medication reported no allergies at the end of the study + total patients who did not receive medication reported no allergies at the end of the study)

$P(M\cap N) = \frac{50}{100}=0.5$

$P(M'\cap N) = \frac{20}{100}=0.2$

Thus, the probability that a patient chosen at random from this study took the medication, given that they reported no allergies,

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

⇒ $P(\frac{M}{N} ) = \frac{0.5}{0.7}=\frac{5}{7}$