Question

Your large corporation manufactures a certain popular brand of robot lawn mower. After manufacture, all mowers go through a standard quality control check. The Portland factory, which manufactures 30% of your mowers, has a probability of .8 that each mower will pass the quality control check. The Dallas factory, which manufactures 50% of your mowers, has a probability of .7 that each mower will pass the quality control check. For the Buffalo factory,
which manufactures the remaining 20% of your mowers, you have been unable to learn what the probability is that each mower will pass inspection. But, the corporation’s annual report claims that the overall probability that one of their manufactured mowers will pass the quality control check is .72

a) What do you conclude is the probability that a mower manufactured at the Buffalo factory will pass the quality control check?
b) A customer orders one of your mowers, and of course receives one that has passed the quality control check. What is the probability it was manufactured in Dallas?

Answer 1

Answer:

(a) The probability that a mower manufactured at the Buffalo factory will pass the quality control check is 0.65.

(b) The probability that a mower was manufactured in Dallas given that it passes the quality check is 0.4861.

Step-by-step explanation:

Denote the events as follows:

X = a mower is manufactured at the Portland factory

Y = a mower is manufactured at the Dallas factory

Z= a mower is manufactured at the Buffalo factory

A = a mower passes the quality check.

The information provided is:

$P(X)=0.30\\P(A|X)=0.80\\P(Y)=0.50\\P(A|Y)=0.70\\P(Z)=0.20\\P(A)=0.72$

(a)

The probability that a mower manufactured at the Buffalo factory will pass the quality control check is:

P (A|Z)

Compute the value of P (A|Z) as follows:

$P(A)=P(A\cap X)+P(A\cap Y) + P (A\cap Z)\\0.72=(0.80\times0.30)+(0.70\times0.50)+(0.20\times P(A|Z))\\0.20\times P(A|Z)=0.72-0.24-0.35\\P(A|Z)=\frac{0.13}{0.20}\\=0.65$

Thus, the probability that a mower manufactured at the Buffalo factory will pass the quality control check is 0.65.

(b)

Compute the value of P (Y|A) as follows:

$P(Y|A)=\frac{P(A|Y)P(Y)}{P(A)}=\frac{0.70\times0.50}{0.72}=0.4861$

Thus, the probability that a mower was manufactured in Dallas given that it passes the quality check is 0.4861.