A bag contains 6 blue balls and 3 green balls and a box contains 4 blue balls…

Concept

When an item is obtained by first randomly choosing one of several containers and then drawing from the chosen one, the Law of Total Probability applies: P(event) = the sum, over every mutually exclusive way of choosing a container, of P(container) times P(event | that container).

Application

1. Since either container is equally likely to be picked, P(bag) = 1/2 and P(box) = 1/2.

2. Bag: 6 blue balls out of 9 total balls, so P(blue | bag) = 6/9.

3. Box: 4 blue balls out of 9 total balls, so P(blue | box) = 4/9.

4. By the Law of Total Probability: P(blue) = 1/2 x (6/9) + 1/2 x (4/9) = 1/2 x (6/9 + 4/9) = 1/2 x (10/9) = 10/18 = 5/9.

Cross-check

Both containers hold 9 balls each and are equally likely to be picked, so pooling directly gives the same result: total blue balls = 6 + 4 = 10, total balls = 9 + 9 = 18, so P(blue) = 10/18 = 5/9, confirming the result.

Answer: 5/9