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Fill rate estimate

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Of $40$ orders sent to a venue, $14$ were filled. Assuming each order is filled independently with the same probability $p$ , which value of $p$ maximises the likelihood?

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Hint

Write the log-likelihood and set its derivative to zero.

Solution

The likelihood is proportional to $p^{14} (1 - p)^{26}$ .
Setting the derivative of the log-likelihood to zero gives $\frac{14}{p} = \frac{26}{1 - p}$ .
The solution is the observed fraction $\frac{14}{40}$ , which reduces to $\frac{7}{20}$ .

Answer: $\frac{7}{20}$

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