Question

question 1
a new drug cures 65% of the patients who take it. if it is given to 14 patients, what is the probability that at least 12 will be cured?
0.0839
0.1512
0.2205
0.0424

Explanation:

Step1: Recall binomial probability formula

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(n\) is the number of trials, \(k\) is the number of successes, \(p\) is the probability of success in a single trial, and \(C(n,k)=\frac{n!}{k!(n - k)!}\)

Here, \(n = 14\), \(p=0.65\), and we want \(P(X\geq12)=P(X = 12)+P(X = 13)+P(X = 14)\)

Step2: Calculate \(P(X = 12)\)

First, calculate \(C(14,12)=\frac{14!}{12!(14 - 12)!}=\frac{14\times13}{2\times1}=91\)
\(P(X = 12)=C(14,12)\times(0.65)^{12}\times(0.35)^{2}\)
\(P(X = 12)=91\times(0.65)^{12}\times(0.35)^{2}\)
\((0.65)^{12}\approx0.005688\), \((0.35)^{2}=0.1225\)
\(P(X = 12)=91\times0.005688\times0.1225\approx0.0643\)

Step3: Calculate \(P(X = 13)\)

\(C(14,13)=\frac{14!}{13!(14 - 13)!}=14\)
\(P(X = 13)=C(14,13)\times(0.65)^{13}\times(0.35)^{1}\)
\((0.65)^{13}\approx0.003697\)
\(P(X = 13)=14\times0.003697\times0.35\approx0.0181\)

Step4: Calculate \(P(X = 14)\)

\(C(14,14)=\frac{14!}{14!(14 - 14)!}=1\)
\(P(X = 14)=(0.65)^{14}\approx0.0015\)

Step5: Sum up the probabilities

\(P(X\geq12)=P(X = 12)+P(X = 13)+P(X = 14)\)
\(P(X\geq12)\approx0.0643 + 0.0181+0.0015=0.0839\)

Answer:

0.0839