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**aaronrj** A small brewery has three bottling machines. Machine A produces 40% of all the bottles, machines B and C produce 30% each. Five percent of bottles filled by A, four percent of bottles filled by B, and three percent of bottles filled by C are rejected for some reason.

(a) If a bottle is filled by A or B, what is the probability that it is rejected?

Consider $\displaystyle 100$ bottles produced by all three machines. Of these, A produces $\displaystyle 40$, B and C $\displaystyle 30$ each. So in $\displaystyle 70$ bottles produced by A and B, $\displaystyle 40$ will come from A and $\displaystyle 30$ from B.

So, given that the bottles come from A and B, the probability that a given bottle is from A is $\displaystyle \tfrac47$, and from B $\displaystyle \tfrac37$. Given that the probabilities that they are rejected are $\displaystyle 0.05$ and $\displaystyle 0.04$ respectively, the probability that a bottle chosen at random is rejected is: $\displaystyle \tfrac47\times 0.05 + \tfrac37\times 0.04 = 0.0457$

Grandad