# Law of Total Probability

• February 4th 2010, 01:23 AM
aaronrj
Law of Total Probability
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?
• February 4th 2010, 08:06 AM
Consider $100$ bottles produced by all three machines. Of these, A produces $40$, B and C $30$ each. So in $70$ bottles produced by A and B, $40$ will come from A and $30$ from B.
So, given that the bottles come from A and B, the probability that a given bottle is from A is $\tfrac47$, and from B $\tfrac37$. Given that the probabilities that they are rejected are $0.05$ and $0.04$ respectively, the probability that a bottle chosen at random is rejected is:
$\tfrac47\times 0.05 + \tfrac37\times 0.04 = 0.0457$