Question

The breakers manufacturing company makes 80% of a particular center, the carton company makes 15% of them, and the flutes company makes the other 5%. The sensors made by breakers have a 4% defect rate, the carton centers have a 6% defect rate, the flute sensors have a 9% defect rate. If a sensor is randomly selected from the general population of all sensors, what is the probability that it is defective, given that it was made by carton Company?

Answer 1

Denote

• $p_1$ - the probability that sensors made by breakers are not defective and $q_1$ - the probability that sensors made by breakers are defective;
• $p_2$ - the probability that sensors made by carton company are not defective and $q_2$ - the probability that sensors made by carton company are defective;
• $p_3$ - the probability that sensors made by flutes company are not defective and $q_3$ - the probability that sensors made by flutes company are defective.

Then

$q_1=0.04,\\p_1=1-0.004=0.96,\\q_2=0.06,\\p_2=1-0.06=0.94,\\q_3=0.09,\\p_3=1-0.09=0.91.$

The probability that selected sensor was made by:

• breakers manufacturing company is 0.8;
• carton company is 0.15;
• flutes company is 0.05.

Use formula for the conditional probability:

$Pr(A|B)=\dfrac{Pr(A\cap B)}{Pr B}=\dfrac{0.8\cdot 0.04+0.15\cdot 0.06+\cdot 0.05\cdot 0.09}{0.15}= 0.3033333\approx 0.3.$

Answer: 0.3, or in percent 30%.

Answer 2

If the randomly selected sensor is identified as having been made by Carton Company, the probability of it being defective must be 0.06, The reason being that it is given that Carton sensors have a 6% defect rate.