Two players with four dollars each after eight tosses

Question

Julie and Judy each begin with $4 and play $8$ rounds of a fair coin toss. On heads Julie pays Judy $1; on tails Judy pays Julie $1. Play only continues while both still have at least $1. What is the probability that after the $8$ tosses each player is back at exactly $4?

Short answer

$17/64$ . They must see exactly $4$ heads and $4$ tails, and nobody may go broke along the way. Among the $\binom84=70$ such orderings only HHHHTTTT and TTTTHHHH bankrupt someone, leaving $68/256=17/64$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

See the hint and the full worked solution in the question bank (free account).

Open the worked solution

Lessons behind this question

Sample Spaces, Events, and the Probability Axioms
Series and the Geometric Series: Partial Sums and Convergence
Conditional Probability, Total Probability, and Bayes' Rule

Start with a free lesson: Probability Fundamentals is part of a free topic. Create a free account, no card needed, to read it in full with worked solutions.

Read it free