Arrivals before the rival stream

MediumProbability Stochastic Processes~5m

Two independent Poisson processes run side by side: stream $A$ at rate $\lambda_A = 1$ per minute and stream $B$ at rate $\lambda_B = 2$ per minute. Watching from time zero, what is the expected number of $A$ -arrivals that occur before the very first $B$ -arrival? (to 3 decimals)

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Merge the streams: each event is independently an $A$ with probability $\lambda_A/(\lambda_A+\lambda_B)$ .
"Number of $A$ 's before the first $B$ " is geometric; its mean is $p/(1-p)$ .

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0.5 (± 0.001)

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Arrivals before the rival stream

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