Please could someone review my solutions for the problems below..thanks in advance
An e-mail message can travel through one of three server routes. The probability of transmission of error in each of the servers and the proportion of messages that travel each route are shown in the following table. Assume that the servers are independent.
% messages % errors Server 1 40 1% Server 2 25 2% Server 3 35 1.5%
1) What is the probability of receiving an email containing an error? Solution: this would be .4*.01 + .25*.02 + .35*0.15 = 0.615
2) What is the probability a msg will arrive without error? Solution: .4*.99 + .25*.98 +.35*.95= .9735
3) If a msg arrives without an error, what is the probability that it was sent through server 2? Solution: Let event E = Sent through server 2. Let event F = arrives without an error. We are looking for P(E/F) or the conditional probability. We can use the formula $\frac {P(E \cap F) }{P(F)} $. P(EnF) = .25*.98 = .245. While the P(F) = 1 - (.01) - (.02) - (.15) = .92. Therefore P(E\F) = .245/.92 = .266 or 26.6%