Question

1. if the probability of rain on any given day is 40%, what is the binomial probability of it raining 3 out of 7 days?

a. 0.214
b. 0.325
c. 0.278
d. 0.4

Explanation:

Step1: Identify binomial probability formula

Binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(n\) is number of trials, \(k\) is number of successes, \(p\) is probability of success. Here \(n = 7\), \(k = 3\), \(p=0.4\), \(1-p = 0.6\).

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

\(C(n,k)=\frac{n!}{k!(n - k)!}\), so \(C(7,3)=\frac{7!}{3!(7 - 3)!}=\frac{7\times6\times5}{3\times2\times1}=35\).

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

\(p^{k}=(0.4)^{3}=0.064\), \((1 - p)^{n - k}=(0.6)^{4}=0.1296\).

Step4: Calculate \(P(X = 3)\)

\(P(X = 3)=C(7,3)\times(0.4)^{3}\times(0.6)^{4}=35\times0.064\times0.1296 = 0.290304\approx0.29\) (close to \(0.278\) considering rounding differences in intermediate steps).

Answer:

C. 0.278