Question

A consumer protection group has gathered data about the performance of a certain brand of refrigerator that has two possible reasons for failure. The data are summarized in the following 2-way table.
mechanical problem

gas shortage Yes No Total
Yes 15 10 25
No 5 70 75
Total 20 80 100
a. What is the probability that a randomly selected refrigerator has gas shortage or mechanical problem?
b. If a randomly selected refrigerator has gas shortage what is the probability that it has mechanical problem?
c. Are the two reasons for failure, gas shortage and mechanical problem independent? Why or why not?

Answer 1

a. $P(Gas\; shortage\;\; OR\;\; Mechanical\; problem)=\frac{25+20-15}{100}=0.3.$

b. $P(Mechanical\;problem|Gas\;shortage)=\frac{15}{25}=0.6.$

c. $P(Gas\;shortage\;\;AND\;\;Mechanical \;problem)=\frac{15}{100}=0.15.$

$P(Gas\;shortage)*P(Mechanical\;problem)=\frac{25}{100}*\frac{20}{100}=0.05.$

Since $0.15\ne0.05,$ gas shortage and mechanical problem are not independent.