Section 6.2

Which Comes First?

Races between events in repeated trials: one total before another, first to k successes, deuce in tennis, two wins in a row against alternating opponents.

Toolkit

Many “race” problems reduce to comparing which of several recurring events happens first, or to counting trials until a run/target is reached. The master tool is first-step (one-step) analysis: condition on the outcome of the next relevant trial and write a recursive equation for the probability or expectation from each state, then solve the resulting linear system.

Key facts:

\mathbb{P}(A\text{ before }B)=\frac{\mathbb{P}(A)}{\mathbb{P}(A)+\mathbb{P}(B)}\quad\text{(conditioning only on trials where $A$ or $B$ occurs)}.

Irrelevant outcomes (favouring neither race) can be discarded; renormalize probabilities among the relevant outcomes.
For symmetric “return to equal” states (deuce, tied scores), a pair of trials either decides the race or returns to the same state, giving a self-consistent equation for the winning probability.
Track progress by a small state (e.g. counts needed of each type), and solve $p_{\text{state}}=\sum (\text{one-step transition})\cdot p_{\text{next state}}$ .
For ordering/scheduling games (two wins in a row, best arrangement of opponents), compare explicit path probabilities; placing the harder task centrally often dominates by direct algebraic comparison.

Problems