Answer to Question #301253 in Statistics and Probability for Uncle B

Question

In a box of switches, it is known that 10% of the switches are faulty. A technician is wiring 30 circuits,

each of which needs one switch. What is the probability that (a) all 30 work (b) at most two of the

switches do not work.

2

Answer 1

Let the random variable $X$ represent the number of faulty switches then, $X\sim Binomial (n=30,p=0.1)$ given as,

$p(X=x)=\binom{30}{x}0.1^x0.9^{30-x}, \space x=0,1,2,3,.....,30$

a)

To find the probability that all 30 switches work is equivalent to finding the probability that there are no faulty switches. This can be written as,

$p(X=0)=\binom{30}{0}0.1^00.9^{30}=0.9^{30}=0.04239116$

The probability that all 30 switches work is 0.04239116

b)

$p(X\le 2)=p(x=0)+p(x=1)+p(x=2)=(\binom{30}{0}0.1^00.9^{30})+(\binom{30}{1}0.1^10.9^{29})+(\binom{30}{2}0.1^20.9^{28})=0.04239116+ 0.1413039+ 0.2276562= 0.4113512$

The probability that there are at most 2 faulty switches is 0.4113512