Two venues publish quote updates as independent Poisson processes at 5 and 15 per second. What is the probability that exactly one of the next four updates comes from the slower venue?

Two venues publish quote updates as independent Poisson processes at 5 and 15 per second. What is the probability that exactly one of the next four updates comes from the slower venue?

Approach: Superposition merges the two streams into one Poisson process at the summed rate, and each merged event independently comes from a given venue with probability equal to that venue's share of the total rate.

27/64. Superposition gives a merged Poisson process at rate 20 per second, and the race between competing exponentials makes each update come from the slower venue with probability 5/20 = 1/4. Those labels are independent of each other and of the arrival times, so the count from the slower venue among the next four is Binomial(4, 1/4), which is binomial thinning of the merged stream. Hence P(exactly one) = C(4,1)(1/4)(3/4)^3 = 4 * 27/256 = 27/64 = 0.4219. The answer carries no reference to a time window, because the independent labels are unrelated to the timing of the events.

Follow-up: What is the expected number of updates from the faster venue before the slower venue publishes its first?

Key concepts: superposition, poisson process, binomial thinning, independent labels.