Question

6. (4.2.12) Of the items manufactured by a certain process, 20% are defective. Of the defective items, 60% can be repaired. a. Find the probability that a randomly chosen item is defective and cannot be repaired. b. Find the probability that exactly 2 of 20 randomly chosen items are defective and cannot be repaired.

Answer 1

(a) Probability that a randomly chosen item is defective and cannot be repaired is 8%.

(b) Probability that exactly 2 of 20 randomly chosen items are defective and cannot be repaired is 0.2711.

Explanation:

We are given that of the items manufactured by a certain process, 20% are defective. Of the defective items, 60% can be repaired.

Let Probability that item are defective = P(D) = 0.20

Also, R = event of item being repaired

Probability of items being repaired from the given defective items = P(R/D) = 0.60

So, Probability of items not being repaired from the given defective items = P(R'/D) = 1 - P(R/D) = 1 - 0.60 = 0.40

(a) Probability that a randomly chosen item is defective and cannot be repaired = Probability of items being defective
`*` Probability of items not being repaired from the given defective items

= 0.20
`*` 0.40 = 0.08 or 8%

So, probability that a randomly chosen item is defective and cannot be repaired is 8%.

(b) Now we have to find the probability that exactly 2 of 20 randomly chosen items are defective and cannot be repaired.

The above situation can be represented through Binomial distribution;

`P(X=r) = \binom{n}{r}p^(r) (1-p)^(n-r) ; x = 0,1,2,3,.....`

where, n = number of trials (samples) taken = 20 items

r = number of success = exactly 2

p = probability of success which in our question is % of randomly

chosen item to be defective and cannot be repaired, i.e; 8%

LET X = Number of items that are defective and cannot be repaired

So, it means X ~
`Binom(n=20, p=0.08)`

Now, Probability that exactly 2 of 20 randomly chosen items are defective and cannot be repaired is given by = P(X = 2)

P(X = 2) =
`\binom{20}{2} * 0.08^(2) * (1-0.08)^(20-2)`

=
`190 * 0.08^(2) * 0.92^(18)`

= 0.2711

Therefore, probability that exactly 2 of 20 randomly chosen items are defective and cannot be repaired is 0.2711.