Permutations and comb A lot consists of 1100 distinct items. There are 4 percent defe

A lot consists of 1100 distinct items. There are 4 percent defective items in the
lot. What is the probability that a random sample of size 50 items contains 4 defective
items?

I did (0.04)^4 x (0.96)^46 *50C4 but I have a feeling it's wrong

Reply 1

old_engineer

Original post by Leah.J

Looks OK to me. 50C4 is the number of different ways the four defective items can be arranged among the sample of 50 items.

Reply 2

divgradzero

The correct answer is (1056C46)(44C4)/(1100C50).

The denominator is 1100C50 as this is the total number of combinations. 4% of 1100 is 44, so there are 44 defective items, and 1056 non-defective in total. There are 44C4 ways of choosing the 4 defective items we need, and 1056C46 ways to choose the remaining (non-defective) items in the sample. Hence, we multiply the two to obtain the numerator.

The answer above is incorrect, as using the Bernoulli trials method assumes that the probability of picking a defective item remains constant. However, this is not the case, as when you pick a defective item, the probability of picking another one drops. For example, at first the probability of picking a defective item is 44/1100 = 0.04, but if we do in fact pick one, then the new probability of picking another one is 43/1099.