Bag A contains 3 red and 5 white balls; Bag B contains 4 red and 4 white balls. A bag is chosen at random and one ball is drawn from it. What is the probability that the ball drawn is red?

Split by which bag is used: (1/2)(3/8) + (1/2)(4/8) = 3/16 + 4/16 = 7/16. Simply adding 3 + 4 red out of 16 balls happens to give the same fraction only because the two bags hold equally many balls, but the correct method is to weight each bag by 1/2.