Question

A lightbulb company has factories in two cities. The factory in city A produces two-thirds of the company’s lightbulbs. The remainder are produced in city B, and of these, 1% are defective. Among all bulbs manufactured by the company, what proportion are not defective and made in city B?

Answer 1

The proportion that is not defective and made in city B is 0.33

Explanation:

Let A be the event "The light bulb does not defect" and B the event "The light bulb is produced in city B".

From the information given:

• The probability of B is
  `P(B)=(1)/(3)`
• The probability of not B is
  `P(B^c)=(2)/(3)`
• The conditional probability of A given B is
  `P(A|B)=0.99`
• The conditional probability of not A given B is
  `P(A^c|B)=1-P(A|B)=0.01`

We want to find the probability of both A and B
`P(A\cap B)`

The conditional probability of events A given B is related by the formula

`P(A|B)=(P(A\cap B))/(P(B))`

Therefore

`P(A \cap B)=P(B)P(A|B)=(0.99)/(3)=0.33`