Question

Plastic parts produced by an injection-molding operation are checked for conformance to specifications. Each tool contains 15 cavities in which parts are produced, and these parts fall into a conveyor when the press opens. An inspector chooses 3 part(s) from among the 15 at random. Three cavities are affected by a temperature malfunction that results in parts that do not conform to specifications. Round your answers to four decimal places.a. what is the probability that the inspector finds exactly one nonconforming part?b. what is the probability that the inspector finds at least one nonconforming part?c. what is the probability that the inspector finds at least two nonconforming part?

Answer 1

Answer:

a) 0.4352

b) 0.5165

c) 0.0813

Step-by-step explanation:

In this problem we have 3 defective parts scattered among 15 parts.

We are going to select 3 out of 15 without replacement, so the situation can be modeled with the Hypergeometric distribution.

If X is the random variable that measures the number of defective parts in a sample of 3, the probability of selecting exactly k defective parts out of 15 would be given by

$P(X=k)=\displaystyle\frac{\binom{3}{k}\binom{15-3}{3-k}}{\binom{15}{3}}$

a) what is the probability that the inspector finds exactly one nonconforming part?

Replacing k with 1 in our previous formula, we get

$P(X=1)=\displaystyle\frac{\binom{3}{1}\binom{15-3}{3-1}}{\binom{15}{3}}=\displaystyle\frac{3*66}{455}=0.4352$

b) what is the probability that the inspector finds at least one nonconforming part?

This would be P(X=1)+P(X=2)+P(X=3) = 1 - P(X=0).  

$P(X=0)=\displaystyle\frac{\binom{3}{0}\binom{15-3}{3-0}}{\binom{15}{3}}=\displaystyle\frac{1*220}{455}=0.4835$

so 1 - P(X=0) = 1 - 0.4835 = 0.5165

c) what is the probability that the inspector finds at least two nonconforming part?

 

P(X=2) + P(X=3) = 1 - (P(X=0) + P(X=1)) = 1 - (0.4835 + 0.4352) =

= 1 - 0.9187 = 0.0813