Question

a medical research study on a new medicine for multiple sclerosis is being conducted with 23 patients. after the study was concluded, it was determined that 18 patients reacted favorably to the medicine, 4 reacted unfavorably, and 1 were unaffected. if three of the patients are randomly selected, determine the probability that none of the patients reacted favorably.
the probability is □.
(type an integer or a simplified fraction.)

Explanation:

Step1: Calculate the number of patients who reacted favorably

The total number of patients is \(n = 23\). The number of patients who reacted unfavorably is \(4\) and the number of patients who were unaffected is \(1\). So the number of patients who reacted favorably is \(23-(4 + 1)=18\).

Step2: Use the combination formula for probability

The formula for the probability of an event \(E\) (in this case, the event that none of the three - selected patients reacted favorably) is \(P(E)=\frac{C(a,r)}{C(n,r)}\), where \(C(n,r)=\frac{n!}{r!(n - r)!}\), \(n = 23\) (total patients), \(r=3\) (number of patients selected), and \(a=23 - 18=5\) (number of non - favorable patients).

First, calculate \(C(5,3)=\frac{5!}{3!(5 - 3)!}=\frac{5!}{3!2!}=\frac{5\times4\times3!}{3!\times2\times1}=10\)

Second, calculate \(C(23,3)=\frac{23!}{3!(23 - 3)!}=\frac{23\times22\times21\times20!}{3!\times20!}=\frac{23\times22\times21}{6}=1771\)

Answer:

\(\frac{10}{1771}\)