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The urn behind a red ball

StatisticsEasy

There are $3$ urns with $4$ balls each. Urn A has $1$ red ball, urn B has $2$ red balls and urn C has $3$ red balls. One urn is chosen uniformly at random and one ball is drawn from it, and the ball is red. What is the probability that urn C was chosen?

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Hint

Weight each urn by its chance of producing a red ball.

Solution

The chances of red are $\frac{1}{4}$ , $\frac{2}{4}$ and $\frac{3}{4}$ for the urns, and the prior is $\frac{1}{3}$ for each.
So the posterior is proportional to $1$ , $2$ and $3$ .
Urn C therefore has probability $\frac{3}{6}$ , which equals $\frac{1}{2}$ .

Answer: $\frac{1}{2}$

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