Q.Three bags contain respectively 3 white & 2 red balls, 7 white & 3 red balls and 5 white & 3 red balls. If a bag is selected at random and one ball is drawn from it, what is the probability that it will be a white ball?

Apply the law of total probability, averaging the white-ball chance over the three equally-likely bags.

Each bag is chosen with probability $\tfrac13$ (a bag is selected at random from three). Given each bag, the probability of drawing white is:

$P(W|\text{Bag1}) = \frac{3}{5},\qquad P(W|\text{Bag2}) = \frac{7}{10},\qquad P(W|\text{Bag3}) = \frac{5}{8}$

By the law of total probability:

$P(W) = \frac13\left(\frac35+\frac{7}{10}+\frac58\right)$

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