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Exact count of the next orders

ProbabilityMedium

A fill rate starts from a uniform $Beta (1, 1)$ prior, and $1$ of $3$ orders fill. What is the posterior predictive chance that exactly $2$ of the next $3$ orders fill? Give a fraction.

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Hint

Find the posterior, then multiply the chances along one order of outcomes and count the orders.

Solution

The posterior is $Beta (2, 3)$ .
A particular order, fill, fill, miss, has chance $\frac{2}{5} \times \frac{3}{6} \times \frac{3}{7} = \frac{3}{35}$ .
Every arrangement has the same chance, and there are $(2 3) = 3$ of them.
The answer is $3 \times \frac{3}{35} = \frac{9}{35}$ .

Answer: $\frac{9}{35}$

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