Section 12.2

Conditioning on How It Ended

Expectations given information about the path: rolls until a 6 given all rolls even (Mossel's paradox), counts of a face before a target, geometric times conditioned on an order.

Toolkit

Conditioning on how a stopping process ended can leave the stopping time’s law unchanged, or drastically reshape it — the key is to identify the exchangeable/symmetric structure underlying the stopping rule.

Core tools:

Symmetry trick: if the distribution of a sum/outcome is symmetric about a value $c$ when restricted to an event, removing a symmetric central slice (e.g. excluding the event itself) preserves the conditional mean at $c$ .
Memorylessness / merging: for independent geometrics $X\sim\mathrm{Geom}(p)$ , $Y\sim\mathrm{Geom}(q)$ , $\min(X,Y)$ is geometric with success probability $p+q-pq$ , independent of which of $X,Y$ “wins” the race.
Enlarging the stopping set: conditioning a geometric waiting time on “no forbidden outcome occurs before the target” is equivalent to redefining success as the union of target and forbidden outcomes; the resulting time is geometric and independent of which element of the enlarged set terminates it.
Order statistics of first-appearance times: among several equally likely recurring symbols, the relative order in which they first appear is a uniformly random permutation.
Conditioning on a count: if $N$ is itself generated by a mixture of “continue” and “stop” steps, conditioning on $N=n$ fixes the mix of step-types used, changing $\mathbb{E}[T\mid N=n]$ linearly in $n$ with slope and intercept read off from the conditional step distribution.

Problems