Question

In a sample of manufacturing process for computer chips 5 out of 100

chips are defective. If 8 chips are chosen at random from a box containing 100 newly manufactured chips, what is the probability (round off to four decimal places)

a) that none of the selected chips are defective?

b) that 4 are non-defective and 4 are defective?

c) that 6 are good and 2 are defective?

d) that at least one is defective?

Answer 1

To determine the probability of different outcomes when selecting computer chips from a batch with a known defect rate, use the hypergeometric distribution formula. The probabilities have been calculated for each scenario, with varying degrees of chips being defective or non-defective.

Step-by-step explanation:

In a manufacturing process for computer chips where 5 out of 100 chips are defective, we can calculate probabilities for different scenarios when choosing 8 chips at random.

Probability of No Defects

For (a), the probability that none of the selected chips are defective can be calculated using the hypergeometric distribution formula. The probability is given by:

P(all non-defective) = (95C8)/(100C8)

Using a calculator, we find P(all non-defective) ≈ 0.7351.

Probability of Half Defective

For (b), the probability that 4 are non-defective and 4 are defective is calculated as:

P(4 non-defective, 4 defective) = (95C4)*(5C4)/(100C8)

Using a calculator, we find P(4 non-defective, 4 defective) ≈ 0.0001.

Probability of Majority Good

For (c), the probability that 6 are good and 2 are defective is calculated as:

P(6 non-defective, 2 defective) = (95C6)*(5C2)/(100C8)

Using a calculator, we find P(6 non-defective, 2 defective) ≈ 0.0746.

Probability of At Least One Defective

For (d), the probability that at least one is defective can be found by subtracting the probability of no defective chips from 1:

P(at least one defective) = 1 - P(all non-defective) ≈ 0.2649.

Answer 2

a) The probability that none of the selected chips are defective is approximately 0.6634.

b) The probability that 4 chips are non-defective and 4 are defective is approximately 0.0038.

c) The probability that 6 chips are good and 2 are defective is approximately 0.2529.

d) The probability that at least one chip is defective is approximately 0.3366.

Step-by-step explanation:

a) To find the probability that none of the selected chips are defective, we need to find the probability of selecting a non-defective chip and multiply it by itself 8 times.

The probability of selecting a non-defective chip is 1 minus the probability of selecting a defective chip.

So, the probability is (95/100) * (95/100) * ... * (95/100) = (95/100)^8.

Solving this expression, we find the probability to be approximately 0.6634.

b) To find the probability that 4 chips are non-defective and 4 are defective, we need to choose 4 non-defective chips from the 95 non-defective chips and 4 defective chips from the 5 defective chips.

The probability of choosing 4 non-defective chips is (95/100) * (94/99) * (93/98) * (92/97), and the probability of choosing 4 defective chips is (5/100) * (4/99) * (3/98) * (2/97).

Multiplying these probabilities, we find the probability to be approximately 0.0038.

c) To find the probability that 6 chips are good and 2 are defective, we need to choose 6 non-defective chips from the 95 non-defective chips and 2 defective chips from the 5 defective chips.

The probability of choosing 6 non-defective chips is (95/100) * (94/99) * (93/98) * (92/97) * (91/96) * (90/95), and the probability of choosing 2 defective chips is (5/100) * (4/99).

Multiplying these probabilities, we find the probability to be approximately 0.2529.

d) To find the probability that at least one chip is defective, we need to find the complement of the probability that none of the chips are defective.

The complement of the probability calculated in part a) is 1 - 0.6634 = 0.3366.