Answer to Question #90689 in Statistics and Probability for Nurul

Question

five defective bulbs are accidently mixed with 20 good ones. it is not possible to just look at a bulb and tell whether or not it is defective. find the probability distribution of the num of defective bulbs, if four bulbs are drawn ar random from this lot answer

Answer 1

Let us denote the number of defective bulbs by X. Clearly X can take the values 0, 1, 2, 3, 4.

P(X = 0) = (no defective bulb) = P(all 4 goods ones) = (20!/(4!*16!))/(25!/(21!*4!))=969/2530

P(X = 1) = P(1 defective and 3 good ones) = (5!/(4!*1!)*20!/(17!*3!))/(25!/(21!/4!))=114/253

P(X = 2) = P(2 defective and 2 good ones) = (5!/(3!*2!)*20!/(18!*2!))/(25!/(21!*4!))=38/253

P(X = 3) = P(3 defective and one good one) = (5!/(3!*2!)*20!/(19!*1!))/(25!/(21!*4!))=4/253

P(X = 4) = P(all 4 defective) = (5!/(4!*1!))/(25!/(21!*4!))=1/2530

So probability distribution is:

x p_i x*p_i

0    969/2530 0

1    114/253      114/253

2       38/253 76/253

3       4/253 12/253

4     1/2530 4/2530

The sum of all x*p_i equals 4/5.