Question

The probability that randomly selected calculator from a store is of brand r is proportional to r, r=1,2,..,6. Further, the probability of a calucltor of brand r being defective is 7 − r 21 , r = 1 , 2 , . . , 6 7 . Then the probability that a calculator randomly selected from the store being defective is A. 8 63 8 B. 13 63 13 C. 55 63 55 D. 50 63 50 Select the correct answer from above options

Answer 1

Correct Answer - A Let C r C denote the event that a calculator of brand r is selected and D r D denote the event that a calculate of brand r is defective. Then, P ( C r ) = λ r and P ( D r / C r ) = 7 − r 21 , r = 1 , 2 , . . , 6 P . Now, 6 ∑ r = 1 P ( C r = 1 ∑ ⇒ 6 ∑ r = 1 λ r = 1 ⇒ λ ( 6 ∑ r = 1 r ) = 1 ⇒ 21 λ = 1 ⇒ λ = 1 21 ⇒ Required probability = 6 ∑ r = 1 P ( C r ∩ D r ) = = 6 ∑ r = 1 P ( C r ) P ( D r / C r ) = = 6 ∑ r = 1 λ r × 7 − r 21 = = λ 21 6 ∑ r = 1 ( 7 r − r 2 ) = = 1 21 2 { 6 ∑ r = 1 7 r − 6 ∑ r = 1 r 2 } = = 1 21 2 { 7 × 21 − 6 × 7 × 13 6 } = 8 63 =