A box contains 3 blue and 2 red marbles while another box contains 2 blue and 5 red marbles. A marble drawn at random from one of the boxes turns out to be blue. What is the probability that it came from the first box?
1 Answer
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Let $R$ (resp. $B$) be event "marble is red (resp. blue)" and $B_k$ be the event "ball came from box number $k$". $$ P(B_1 \mid B) = \frac{P(B \mid B_1)P(B_1)}{P(B)},\; P(B) = P(B \mid B_1)P(B_1) + P(B \mid B_2)P(B_2) $$ where the first equality is Bayes rule, and the second is the so-called formula for "total probability".
You have all the numbers $P(B_1)$, $P(B_2)$, $P(B \mid B_1)$, $P(B \mid B_2)$ given in the statement of your problem. Can your proceed from here ?
