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Which stream is larger

StatisticsHard

Independent Poisson streams count events in the same hour: stream A has mean $1$ and stream B has mean $2$ . Given that the total count is $6$ , what is the probability that stream B produced more events than stream A?

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Hint

Given the total, the count from one stream is binomial.

Solution

Given a total of $6$ , the number of events from stream A is binomial with $n = 6$ and $p = \frac{1}{3}$ .
Stream B has more events exactly when A has $0$ , $1$ or $2$ events.
The probability is $\frac{64 + 192 + 240}{729}$ , which equals $\frac{496}{729}$ .

Answer: $\frac{496}{729}$

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