Answer to Question #91890 in Statistics and Probability for Nancy

Question

An aircraft emergency locator transmitter (ELT) is a device designed to transmit a
signal in the case of a crash. The Altigauge Manufacturing Company makes 80% of
the ELTs,the Bryant Company makes 15% of them, and the Chartair Company makes
the other 5%. The ELTs made by Altigauge have a 4% rate of defects, the Bryant
ELTs have a 6% rate of defects, and the Chartair ELTs have a 9% rate of defects
(which helps to explain why Chartair has the lowest market share).
a. If an ELT is randomly selected from the general population of all ELTs, find
the probability that it was made by the Altigauge Manufacturing Company.
b. If a randomly selected ELT is then tested and is found to be defective, find the
probability that it was made by the Altigauge Manufacturing Company

Answer 1

Let event A = ELT is made by  the Altigauge Manufacturing Company,

event B = ELT is made by ,the Bryant Company,

event C= ELT is made by the Chartair Company,

event D = ELT has a defect.

Since  the Altigauge Manufacturing Company makes 80 % of ELTs P(A) = 0.80,

the Bryant Company makes 15% of ELTs, therefore P(B) = 0.15,

the Chartair Company makes  of ELTs, therefore P(C) = 0.05.

The ELTs made by Altigauge have a 4% rate of defects, the Bryant

ELTs have a 6% rate of defects, and the Chartair ELTs have a 9% rate of defects, hence

conditional probabilities P(D|A) = 0.04, P(D|B) = 0.06, P(D|C) = 0.09.

By Bayes' theorem conditional probability

$$P(A|D)=\frac{P(D|A)P(A)}{P(D|A)P(A)+P(D|B)P(B)+P(D|C)P(C)}=$$

$$\frac{0.04*0.8}{0.04*0.8+0.06*0.15+0.09*0.05}=64/91=0.\overline{703296}$$

Answer: a) 0.8, b) 64/91.