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A rookie's posterior win rate

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A trader's win rate has prior $Beta (6, 4)$ . In the first $5$ independent trades the trader wins $4$ . What is the posterior mean of the win rate? Give a fraction.

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Hint

A Beta prior acts like wins and losses already seen.

Solution

The posterior is $Beta (6 + 4, 4 + 1)$ , which is $Beta (10, 5)$ .
Its mean is $\frac{10}{10 + 5} = \frac{10}{15}$ .
So the posterior mean is $\frac{2}{3}$ .

Answer: $\frac{2}{3}$

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