Question

B2. A package of 6 fuses are tested where the probability an individual fuse is defective is
0.05. (That is, 5% of all fuses manufactured are defective).
(a) What is the probability one fuse will be defective?
(b) What is the probability at least one fuse will be defective?
(c) What is the probability that more than one fuse will be defective, given that at
least one is defective?

Answer 1

This is a binomial distribution case where p=0.05,n=6, q=1-0.05=0.95

Let X denote the random variable for the number of fuses.

$P(X=x) ={n\choose x} p^x q^{n-x}$

a) p(X=1)

$={6\choose 1}{0.05^1}{0.95^5}$

=0.2321

b) $P(X \geq 1)$

$={\sum_{x=1}^6} 0.05^x 0.95^{6-x}$

=0.2649

c)$P(X>1|X\geq 1)=\frac {p(x>1)}{p(X\geq 1}$

P(X>1)$={\sum_{x=2}^6} 0.05^x 0.95^{6-x}$

=0.0327

$P(X>1|X\geq 1)=\frac{0.0327}{0.2649}$

=0.1237