Question

"A shipment has 20 items. Five items will be picked randomly and checked for quality. If at least one item is defective, then the shipment is rejected. Given that the shipment has k defective items, what is the probability that it is rejected?"

Answer 1

Answer:

$1 - \frac{(20-k)(19-k)(18-k)(17-k)(16-k)}{20*19*18*17*16}$

Step-by-step explanation:

If k items out of 20 total items are defective, then the number of non-defective item is 20 – k. The probability of choosing 5 items where all of them are non-defective is

- For the 1st slot the chance is (20 – k)/20

- For the 2nd slot the chance is (20-k-1)/(20-1) = (19-k)/19

- For the 3rd slot: (18 – k)/18

- 4th : (17 – k)/17

- 5th: (16 – k)/16

The probability in total would be

$\frac{(20-k)(19-k)(18-k)(17-k)(16-k)}{20*19*18*17*16}$

So the probability of selecting at least 1 defective item is the inverse of this, which is

$1 - \frac{(20- k)(19-k)(18-k)(17-k)(16-k)}{20*19*18*17*16}$