Question

intext:"A shipment of 50 inexpensive digital​ watches, including 6 that are​ defective, is sent to a department store. The receiving department selects 10 at random for testing and rejects the whole shipment if 1 or more in the sample are found defective. What is the probability that the shipment will be​ rejected?"

Answer 1

Answer:

0.7125

Step-by-step explanation:

The binomial distribution with parameters n and p is the discrete probability distribution of the number of successes (with probability p) in a sequence of n independent events.

The probability of getting exactly x successes in n independent Bernoulli trials =  $n_{C_{x}}(p)^x(1-p)^{n-x}$

Total number of watches in the shipment = 50

Number of defective watches = 6

Number of selected watches = 10

Let X denotes the number of defective digital watches such that the random variable X follows a binomial distribution with parameters n and p.

So,

Probability of defective watches = $\frac{X}{n}=\frac{6}{50}=0.12$

Take n = 10 and p = 0.12

Probability that the shipment will be rejected = $P(X\geq 1)=1-P(X=0)$

$=1-n_{C_{x}}(p)^x(1-p)^{n-x}\\=1-10_{C_{0}}(0.12)^0(1-0.12)^{10-0}$

Use $n_{C_{x}}=\frac{n!}{x!(n-x)!}$

So,

Probability that the shipment will be rejected = $=1-\left ( \frac{10!}{0!(10-0)!} \right )(0.88)^{10}$

$=1-(0.88)^{10}\\=1-0.2785\\=0.7125$