Question

When a consignment of pens arrives at the retailer's, ten of them are tested. The whole batch is returned to the wholesaler if more than one of those selected is found to be faulty. What is the probability that the consignment will be accepted if 2% of the pens are faulty?

Answer 1

To find the probability that the consignment will be accepted, calculate the probability of selecting no more than one faulty pen out of ten selected pens.

Step-by-step explanation:

To find the probability that the consignment will be accepted, we need to find the probability that no more than one pen is faulty. We know that 2% of the pens are faulty. Let's calculate the probability step by step:

1. Calculate the probability of selecting a faulty pen: 2% = 0.02
2. Calculate the probability of selecting a non-faulty pen: 1 - 0.02 = 0.98
3. Calculate the probability of selecting no more than one faulty pen out of ten selected pens:

• P(selecting 0 faulty pens) = (0.98)^10 = 0.817
• P(selecting 1 faulty pen) = 10 * (0.98)^9 * (0.02) = 0.169
• P(no more than one faulty pen) = P(selecting 0 faulty pens) + P(selecting 1 faulty pen) = 0.817 + 0.169 = 0.986

Therefore, the probability that the consignment will be accepted is 0.986, or 98.6%.

Answer 2

To calculate the probability that a consignment of pens with a 2% fault rate will be accepted upon testing 10 pens, you would sum the binomial probabilities of observing exactly 0 and exactly 1 faulty pen.

Step-by-step explanation:

With the given fault rate (2%, or 0.02), we can calculate the probability that 0 pens are faulty and the probability that 1 pen is faulty, then sum these probabilities to find the overall chance the consignment is accepted.

Calculating '0 faulty pens' probability:

For all ten pens to be non-faulty, we use the binomial probability formula with p=0.02 (the probability of a pen being faulty), n=10 (the number of pens), and x=0 (the desired number of faulty pens):

P(X=0) = (n choose x) * p^x * (1-p)^(n-x)

Which simplifies to: P(X=0) = (10 choose 0) * 0.02^0 * (1-0.02)^10

Calculating '1 faulty pen' probability:

For exactly one pen to be faulty, we change x to 1 in the binomial formula:

P(X=1) = (n choose x) * p^x * (1-p)^(n-x)

Which simplifies to: P(X=1) = (10 choose 1) * 0.02^1 * (1-0.02)^9

Finally, we add the probabilities of these two scenarios to find the overall acceptance probability: Probability of acceptance = P(X=0) + P(X=1).

Conclusion

The final probability of acceptance is the sum of the probabilities of exactly 0 and exactly 1 pen being faulty in the randomly selected sample of 10.