Question

An aircraft emergency locator transmitter is a device designed to transmit a signal in the case of a crash. The Alpha Manufacturing Company makes 75% of the transmitters, the Beta Company makes 20% of them, and the Gamma Company makes the other 5%. The transmitters made by Alpha have a 4% rate of defects, the Beta’s transmitters have a 6% rate of defects, and the Gamma’s transmitters have a 9% rate of defects (which helps to explain why Gamma has the lowest market share).

1. If a randomly selected transmitter is tested and is found to be defective, find the probability that it was made by the Alpha Manufacturing Company.

2. If a randomly selected transmitter is tested and is found to be non-defective, What is the probability it was not made by Gamma Company?
3. Is the choice of the manufacturing company independent of the quality (defective or not) of the transmitters? justify

Answer 1

a)

P(defective)=P(made by Alpha and defective)+P(made by BEta and defective)+P(made by gamma and defective)=0.75*0.04+0.2*0.06+0.05*0.09=0.0465

hence P(Alpha given defective)==P(made by Alpha and defective)/P(defective)

=0.75*0.04/0.0465=0.645161

b)

P(not by Gamma given not defective)=1-P(by Gamma given not defective)

=1-P(made by gamma and defective)/P(not defective)=1-0.05*0.91/(1-0.0465)=0.952281

c)

as P(Alpha given defective) is not equal to P(Alpha) ; therefore choice of the manufacturing company is not independent of the quality of the transmitters