Question

A manufacturing process produces semiconductor chips with a known failure rate of 6.3%. Assume that the chip failures are independent of one another. You will be producing 2,000 chips tomorrow.

e. Find the probability that you will produce more than 120 defects.

f. You just learned that you will need to ship 1,860 working chips out of tomorrow’s production of 2,000. What are the chances that you will succeed? Will you need to increase the scheduled number produced?

g. If you schedule 2,100 chips for production, what is the probability that you will be able to ship 1,860 working ones?

Answer 1

A manufacturing process produces semiconductor chips with a known failure rate of 6.3%. Assume that the chip failures are independent of one another. You will be producing 2,000 chips tomorrow.

e. Find the probability that you will produce more than 120 defects.

f. You just learned that you will need to ship 1,860 working chips out of tomorrow’s production of 2,000. What are the chances that you will succeed? Will you need to increase the scheduled number produced?

g. If you schedule 2,100 chips for production, what is the probability that you will be able to ship 1,860 working ones?

Answer

The manufacturing process produces semiconductor chips with a known failure rate of 6.3% and we assume that the chip failures are independent of one another.

Let, X is the random variable denoting the number of working chips out of 2000 chips that will be produced tomorrow.

Since, the failure rate of chip production is 6.3%, the success rate = (100 - 6.3)% = 93.7% = 0.937, which is the probability of success.

As the chip failures are independent of one another, we have

, X = 0(1)2000

The probability mass function of X is given by,

e. The probability that we will produce more than 120 defects

= The probability that we will produce less than (2000 - 120) = 1880 working chips

= 1 - 0.3099 (rounded to 4 decimal places)

= 0.6901

Answer: The probability that we will produce more than 120 defects is 0.6901.

f. If we have to ship 1860 working chips out of the production of 2000, the chances of our success is

= 0.0157 (rounded to 4 decimal places)

Answer: If we have to produce 1,860 working chips out of the production of 2,000, the chances of our sucess is 0.0157. Since, the probability is very low I think we need to increase the scheduled number produced.

g. If we schedule 2,100 chips for production, the distribution of X will change to

, X = 0(1)2100

and the probability mass function of X will be given by,

Then the probability that we will be able to ship 1860 working chips

Answer: If you schedule 2,100 chips for production, the probability that we will be able to ship 1,860 working ones is .