Question

Maine Corporation buys motors for electric fans from another company that guarantees that at most 5% of its motors are defective and that it will replace all defective motors at no cost to Maine Corporation. The motors are received in large shipments. The quality control department at Maine Corporation randomly selects 20 motors from each shipment and inspects them for being good or defective. If this sample contains more than two defective motors, the entire shipment is rejected.

Using the appropriate probabilities table from Appendix B, find the probability that a given shipment of motors received by Maine Corporation will be accepted. Assume that 5% of all motors received are defective.
Using the appropriate probabilities table from Appendix B, find the probability that a given shipment of motors received by Maine Corporation will be rejected.

Answer 1

Accepted: 0.9246 Rejected: 0.0754

Explanation:

To find the probability that a given shipment of motors received by Maine Corporation will be accepted, we need to calculate the probability of having at most two defective motors in a sample of 20 motors.

Given that 5% of all motors received are defective, we can assume that the probability of selecting a defective motor is 0.05, and the probability of selecting a good motor is 0.95.

To calculate the probability of having at most two defective motors, we need to calculate the probability of having 0, 1, or 2 defective motors and then sum them up.

Let's calculate each probability step by step:

1. Probability of having 0 defective motors:

P(0 defective motors) = (0.95)^20

This is because we need to select 20 good motors in a row.

P(0 defective motors)

0.3585

2. Probability of having 1 defective motor:

P(1 defective motor) = 20C1 * (0.05)^1 * (0.95)^19

This is because we need to select 1 defective motor and 19 good motors in any order.

P(1 defective motor)

0.3774

3. Probability of having 2 defective motors:

P(2 defective motors) = 20C2 * (0.05)^2 * (0.95)^18

This is because we need to select 2 defective motors and 18 good motors in any order.

P(2 defective motors)

0.1887

Now, we can sum up these probabilities to find the probability of accepting the shipment:

P(accepted) = P(0 defective motors) + P(1 defective motor) + P(2 defective motors)

P(accepted)

0.3585 + 0.3774 + 0.1887

P(accepted)

0.9246

Therefore, the probability that a given shipment of motors received by Maine Corporation will be accepted is approximately 0.9246.

To find the probability that a given shipment of motors received by Maine Corporation will be rejected, we can subtract the probability of acceptance from 1:

P(rejected) = 1 - P(accepted)

P(rejected)

1 - 0.9246

P(rejected)

0.0754

Therefore, the probability that a given shipment of motors received by Maine Corporation will be rejected is approximately 0.0754.