Waiting for the Fill

Question

Your resting limit order has an independent 30% chance of being filled during each second it sits in the book. What is the probability it gets filled within the first five seconds?

Accepted Answer

The order survives five seconds unfilled only by missing five independent 30% chances: P(textunfilled after 5s) = 0.7^5 = 0.16807, P(textfilled within 5s) = 1 - 0.7^5 approx 0.832. This is the geometric distribution's CDF: P(T le n) = 1 - (1-p)^n. The trap Adding 5 times 0.3 = 1.5 is not a probability, per-period probabilities of union events never add unless the events are disjoint. The complement trick is the only safe general route. Two properties worth saying out loud Memorylessness: if the order is still unfilled after three seconds, the probability it fills in the next five is still 0.832, the geometric clock carries no aging. Median vs mean: the median fill time is 2 seconds (1 - 0.7^2 = 0.51 > frac12) while the mean is frac10.3 approx 3.33, the right-skewed tail drags the mean above the median. Queue-position and fill-probability reasoning at market-making firms is built on exactly this model, usually in its continuous (exponential/Poisson) form.

Waiting for the Fill

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