Section 6.1

Taking Turns

Players alternate attempts until the first success (dice, shooters, re-spun revolvers): first-step analysis or a geometric series.

Toolkit

When players take turns attempting a success, condition on the outcome of the first round of attempts: either someone succeeds, settling the game, or everyone fails, in which case the game restarts from scratch with identical odds. This self-similarity lets you write an equation for the quantity of interest in terms of itself, rather than summing an infinite series directly.

For player order with success probabilities $p_1,p_2,\dots,p_k$ acting in turn each round, let $q_i=1-p_i$ . The probability the first player eventually wins satisfies

\mathbb{P}(\text{player }1\text{ wins}) = \frac{p_1}{1-q_1q_2\cdots q_k},

since each player’s winning probability is their own first-round success chance divided by the probability that the round is not a total failure for everyone. More generally, player $i$ ’s win probability is

\frac{q_1\cdots q_{i-1}\,p_i}{1-q_1 q_2\cdots q_k}.

Expected number of rounds to end the game is $1/(1-q_1\cdots q_k)$ .

Problems

6.1 Alternating Until First Success+5 variations Easy Free