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Evidence over a pair of days

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A signal hits $3$ of $4$ independent trades on Monday and $2$ of $4$ on Tuesday. What is the likelihood ratio over both days for hit rate $\frac{3}{4}$ against hit rate $\frac{1}{2}$ , with $\frac{3}{4}$ on top? Give a fraction.

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Hint

Independent days multiply, so each hit and each miss contributes one factor.

Solution

A hit multiplies the ratio by $\frac{\frac{3}{4}}{\frac{1}{2}} = \frac{3}{2}$ , a miss by $\frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2}$ .
Over both days there are $3 + 2 = 5$ hits and $8 - 5 = 3$ misses.
The ratio is $(\frac{3}{2})^{5} \times (\frac{1}{2})^{3} = \frac{243}{256}$ .

Answer: $\frac{243}{256}$

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