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Quote updates follow a Poisson process with a constant rate. You are told that exactly $5$ updates arrived in a $10$ -minute window. What is the probability that exactly $2$ of them arrived in its first $2$ minutes? Give a fraction in lowest terms.

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Hint

Given the count, arrival times behave like independent uniform points.

Solution

Each update lands in the first $2$ minutes with probability $\frac{2}{10} = \frac{1}{5}$ , whatever the rate.
The count there is $Binomial (5, \frac{1}{5})$ , so the chance is $(2 5) \times (\frac{1}{5})^{2} \times (\frac{4}{5})^{3} = \frac{128}{625}$ .

Answer: $\frac{128}{625}$

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