A lot of 100 items contains? defective items.? items are chosen at random and tested. what is the probability that? are found defective?

A lot of 100 items contains k defective items. n items are chosen at random and tested. what is the probability that x are found defective?

Define x, n and p as above.

Use hypergeometric distribution where:

a=number of defective items selected

A=total number of defective items

b=number of non-defective items selected

B=total number of non-defective items

Then

P (a, b) = C (A, a) C (B, b) / C (A+B, a+b)

where

C (n, r) = n! / (n! (n-r) !) = combination of r items selected from n,

A+B=total number of items

a+b=number of items selected

Then

A=k

B=100-k

a=x

b=n-x n ≥ x

Substitute in hypergeometric distribution, we have

P (x) = C (A, a) C (B, b) / C (A+B, a+b)

=C (k, x) C (100-k, n-x) / (C (100, n)

recall:

C (n, r) = n! / (n! (n-r) !) = combination of r items selected from n,