Question

In a box of 12 light bulbs, it is known that 5 of them are defective. That is, 5 are defective and 7 function properly. We select 2 bulbs from the bag at random, without replacement. Find the probability they both function properly. Show work. Round your answer to 3 decimal places.

Answer 1

Solution:

Given:

We are given that: in a box of 12 light bulbs, it is known that 5 of them are defective.

Thus Number of defective = 5 and Number of functioning properly = 12-5=7

2 bulbs are selected from the bag at random, without replacement.

We have to find: the probability that they both function properly.

That is we have to find:

P(X = 2) = .......?

X : Number of bulbs function properly

M = Number of bulbs function properly = 7

N = Total Bulbs = 12

n = sample size selected = 2

x = number of bulbs from sample which are function properly

Thus X follows Hypergeometric distribution with parameters N = 12 , M = 7 and n = 2

Thus probability mass function of Hypergeometric distribution is:

where

Thus we get:

We can find same answer using another steps:

Since bulbs are selected without replacement, for first draw we have total 12 bulbs and 7 are function properly

thus P( first bulb function properly) =7/12

For second draw we have total bulbs = 12-1 = 11 total and 7-1=6 function properly

thus

P( Second bulb function properly) = 6 / 11

Thus

P( Both function properly ) =7/12 X 6/11

P( Both function properly ) = 0.583333 X 0.545455

P( Both function properly ) = 0.318182

P( Both function properly ) = 0.318