# Total probability, Bayes

There are 2 identical containers. The first one has 5 white and 8 black balls, the second one has 7 white and 9 black balls. 3 balls are randomly taken out of the first container and 4 balls are randomly taken from the second container. All of the balls (7) are put in a third container. If a ball is taken out of the third container, what is the probability that it will be white?

## 1 Answer

The event "ball taken out at the end is white" can happen in two disjoint ways: (i) The ball came originally from the first container, and is white or (ii) The ball came originally from the second container, and is white.

The probability that the chosen ball came from the first container is $\frac{3}{7}$. Given that it came from the first container, the probability that it is white is $\frac{5}{13}$. So the probability it came from the first container and is white is $\frac{3}{7}\cdot \frac{5}{13}$.

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