Question

in binomial probability, if the probability of success is 0.5 and the number of trials is 4, what is the probability of exactly 2 successes?
a. 0.25
b. 0.625
c. 0.5
d. 0.375

Explanation:

Step1: Use 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, and \(C(n,k)=\frac{n!}{k!(n - k)!}\). Here, \(n = 4\), \(k = 2\), \(p=0.5\), and \(1-p = 0.5\).

Step2: Calculate combination \(C(n,k)\)

\(C(4,2)=\frac{4!}{2!(4 - 2)!}=\frac{4!}{2!×2!}=\frac{4\times3\times2!}{2!×2!}=\frac{4\times3}{2\times1}=6\)

Step3: Calculate \(p^{k}\times(1 - p)^{n - k}\)

\(p^{k}\times(1 - p)^{n - k}=(0.5)^{2}\times(0.5)^{4 - 2}=(0.5)^{2}\times(0.5)^{2}=(0.5)^{4}=\frac{1}{16}\)

Step4: Calculate \(P(X = 2)\)

\(P(X = 2)=C(4,2)\times p^{k}\times(1 - p)^{n - k}=6\times\frac{1}{16}=\frac{6}{16}=0.375\)

Answer:

d. \(0.375\)