Answer to Question #242376 in Statistics and Probability for GAD

Question

Box 1 contains 1000 bulbs of which 10% are defective. Box 2 contains 2000 bulbs of which 5%

are defective. Two bulbs are drawn (without replacement) from a randomly selected box. (i) Find the probability that both bulbs are defective, and

(ii) Assuming that both are defective, find the probability that they came from box 1.

Answer 1

Let B1 represent box 1 and B2 represent box 2.Defective bulbs are represented by D and N for non-defective bulbs.

Number of defective bulbs from box1 is 10% which can be given as,

Defective bulbs=10/100*1000=100

Number of defective bulbs from box2 is 5% which can be given as,

Defective bulbs =5/100*2000 =100

Probability for randomly selecting a box is given by,

$p(B1)=p(B2)=1/2$

Probability for selecting a defective bulb given box 1 is,

$p(D/B1)$=100/1000*99/999=99/9990=0.0099(4 decimal places)

and given box 2 is,

$p(D/B2)$=100/2000*99/1999=99/39980=0.0025(4 decimal places)

(i).

To get the probability that both bulbs are defective, the law of total probability is used as illustrated below.

$p(D)=p(D/B1)*p(B1)+p(D/B2)*p(B2)$

=(0.0099)*1/2+(0.0025)*1/2

=0.0062(4 decimal places)

Probability that both bulbs are defective is 0.0062.

(ii).

The conditional probability determined in this part is p(B1/D) and is given by,

$p(B1/D)=p(D/B1)*p(B1)/p(D)$

=(0.0099)*1/2/0.0062

= 0.004954955/0.0062

=0.80(2 decimal places)

Probability that bulbs came from box 1 given that both bulbs are defective is 0.8