Trades arrive as a Poisson process at 12 per minute. Independently of everything else, each trade is large with probability 1/4 and small otherwise. What is the probability that a given minute contains exactly 2 large trades and at least 1 small trade?

Trades arrive as a Poisson process at 12 per minute. Independently of everything else, each trade is large with probability 1/4 and small otherwise. What is the probability that a given minute contains exactly 2 large trades and at least 1 small trade?

Approach: Thinning splits a Poisson process into independent Poisson processes carrying the split rates, so the two counts factorise and each can be handled on its own.

0.2240. Poisson thinning makes large trades a Poisson process at rate 3 per minute and small trades an independent Poisson process at rate 9. The independence of split streams lets the answer factorise: [e^{-3} 3^2/2!] * [1 - e^{-9}] = 0.22404 * 0.99988 = 0.2240. Independence is the substantive claim, since knowing there were 2 large trades says nothing about the small ones. That would fail if the total count were fixed, where the split is binomial and the two counts are negatively related. Superposition runs the argument backwards, merging the two streams into a Poisson process at rate 12.

Follow-up: Given that a minute contained 12 trades in total, what is the probability exactly 2 of them were large?

Key concepts: poisson thinning, superposition, independence of split streams, poisson process.