A randomly selected purchaser has a router that needs repair under warranty. what is the probability that the router is brand 1?

Question

A store sells four brands of routers. The least expensive brand 1 accounts for 40% of the sales. The other brands (in order of their price) have the following percentages of sales: 2, 30%; 3, 20%; and 4, 10%. The respective probabilities of needing repair during warranty are 0.1, 0.05, 0.03, and 0.02, for brands 1, 2, 3, and 4.

A randomly selected purchaser has a router that needs repair under warranty. what is the probability that the router is brand 1?

do I just multiply probability of B1 sales and needing repair together? (0.4 x 0.1)

Answer 1

HINT

With $B_i$ = Brand i, and $R$ = needing repair ,

you need to find $P(B_i|R) = \dfrac{P(B_i)*P(R|B_i)}{\Sigma[ P(B_i)*P(R|B_i)]}$

FURTHER HINT

A commonsense way which in effect is using Bayes' Rule is to assume that, say $1000$ pieces have been manufactured, and compute numbers needing repair for each brand

Brand $B_1$: $400$ pieces, $40$ needing repair
Brand $B_2$: $300$ pieces, $15$ needing repair,
and so on

What fraction of pieces needing repair are from brand $B_1$ ?

Using the formula given in the first part, answer will be

$\dfrac{0.4*0.1}{0.4*0.1+0.3*0.05+ 0.2*0.03+0.1*0.02}$

Answer 2

Going by the formula provided by in true blue anil in his answer, I believe the probability should be $0.63$.