Answer to Question #90689 in Statistics and Probability for Nurul

Question #90689
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
1
Expert's answer
2019-06-11T09:23:16-0400

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.


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