Machines and manufacture identical bolts and contribute and of the total daily output

MATHEMATICS · Multi-part · CBSE · Mathematics · Probability · Probability

Problem

Machines

A, B,

and

C

manufacture identical bolts and contribute

30\%, 20\%,

and

50\%

of the total daily output, respectively. Their defective rates are

2\%

for

A

3\%

for

B

, and

4\%

for

C

. A bolt is picked at random from the day's production. A compensation amount

X

(in rupees) is defined as follows: • If the selected bolt is non-defective,

X=0

. • If the bolt is defective and produced by

A

X=5

. • If the bolt is defective and produced by

B

X=7

. • If the bolt is defective and produced by

C

X=9

. Answer the following questions: (i) Write down the unconditional probability distribution of

X

. (ii) Find the unconditional expected compensation

E(X)

. (iii) Given that the selected bolt is defective, obtain the conditional probability distribution of

X

. (iv) Hence calculate

E(X\,|\,\text{bolt is defective})

and compare it with the value in part (ii).