Probability

Concept:Use the law of total probability to combine the probabilities from each box.Explanation:Let Box 1 have 5 white and 6 blue balls, total 11.Probability of blue from Box 1 is $\frac{6}{11}$.Box 2 has 3 white and 7 blue balls, total 10.Probability of blue from Box 2 is $\frac{7}{10}$.The box is chosen at random, so each box has probability $\frac{1}{2}$.Total probability that the ball is blue:$$P(\text{blue}) = \frac{1}{2} \cdot \frac{6}{11} + \frac{1}{2} \cdot \frac{7}{10} = \frac{1}{2} \left( \frac{6}{11} + \frac{7}{10} \right)$$Compute $\frac{6}{11} + \frac{7}{10} = \frac{60+77}{110} = \frac{137}{110}$.Then $P(\text{blue}) = \frac{1}{2} \cdot \frac{137}{110} = \frac{137}{220}$.Answer:$\frac{137}{220}$