Question

at a manufacturing plant where switches are made, it is a known fact that 2% of all switches are defective. if two switches are used in a device, what is the probability that exactly one switch is good?
a. 0.9604
b. 0.0392
c. 0.0196
d. 2
please select the best answer from the choices provided

Explanation:

Step1: Calculate the probability of a good switch

The probability that a switch is defective \(p = 0.02\), so the probability that a switch is good \(q=1 - p=1 - 0.02 = 0.98\)

Step2: Use the binomial probability formula

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times q^{n - k}\), where \(n\) is the number of trials, \(k\) is the number of “successes” (in this case, the number of defective switches), \(C(n,k)=\frac{n!}{k!(n - k)!}\)

Here \(n = 2\) (two switches), \(k = 1\) (exactly one defective switch, which means exactly one good switch).

First, calculate \(C(2,1)=\frac{2!}{1!(2 - 1)!}=\frac{2!}{1!1!}=2\)

Then \(p = 0.02\), \(q = 0.98\), \(n = 2\), \(k = 1\)

\(P(X = 1)=C(2,1)\times(0.02)^{1}\times(0.98)^{2 - 1}\)

\(P(X = 1)=2\times0.02\times0.98\)

\(P(X = 1)=0.0392\)

Answer:

B. 0.0392