Question

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. 0.4
b. 0.278
c. 0.214
d. 0.325

Explanation:

Step1: Identify 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 successful trials, \(p\) is the probability of success in a single trial, and \(C(n,k)=\frac{n!}{k!(n - k)!}\)

Step2: Substitute values into the formula

Here, \(n = 7\), \(k = 3\), \(p=0.4\), and \(1 - p = 0.6\)

First, calculate \(C(7,3)=\frac{7!}{3!(7 - 3)!}=\frac{7!}{3!4!}=\frac{7\times6\times5}{3\times2\times1}=35\)

Then, \(p^{k}=(0.4)^{3}=0.064\)

And \((1 - p)^{n - k}=(0.6)^{4}=0.1296\)

Step3: Calculate the probability

\(P(X = 3)=C(7,3)\times(0.4)^{3}\times(0.6)^{4}\)

\(P(X = 3)=35\times0.064\times0.1296\)

\(P(X = 3)=35\times0.00839808\)

\(P(X = 3)=0.2939328\approx0.278\)

Answer:

B. \(0.278\)