Question

Among the computer chips produced at a computer facility, 2% are mildly defective and an additional 3% are highly defective. The remaining chips are OK. Different chips are independent in terms of status. If a chips is mildly defective, it costs $20 to repair it to proper condition. For highly defective chips, the cost is $60. There is no repair cost for OK chips. Suppose that 40 chips produced at this facility are examined for defects: 1. What is the probability that exactly two chips are defective up to some extent? (5 points) 2. What is the probability that exactly two chips are defective and they are examined consecutively? (for example third and fourth chips are defective)(5 points) 3. For the first chip examined, what is the mean and variance of the repair cost? (5 points) 4. What is the probability that the total repair cost for the 40 chips is exactly $60? (5 points)

Answer 1

In total 5% of the chips are defective to some extent.

1) The probability that exactly 2 chips are defective to some extent is

2. The number of ways in which two consecutive chips can be selected from 40

= 40 - 1 = 39

The probability that exactly 2 chips are defective and they are examined consecutively is

3. The probability that the first chip is mildly defective is = 0.02

The probability that the first chip is highly defective = 0.03

The mean repair cost for the first chip is

= 0.02 * $20 + 0.03 * $60

= $2.2

The variance of the repair cost for the first chip

= $20 * 0.02 * (1-0.02) + $60 * 0.03 * (1-0.03)

= 0.392 + 1.746

= 2.138 dollar^2

4. The expected repair cost for the 40 chips is = $2.2 * 40 = $88

The variance of the repair cost for the 40 chips is = 2.138 * 40 = 85.52

The standard deviation of the repair cost for the 40 chips is = $9.248

The probability that the repair cost will be exactly $60 can be computed using the continuity correction.

The probability that the total repair cost will be exactly $60 is = 0.0005

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