One extra coin and the chance of more heads

Question

Emilia tosses $n+1$ fair coins while Ron tosses $n$ . Find the chance that Emilia's head count strictly exceeds Ron's.

Short answer

$1/2$ . Compare Ron's $n$ coins with Emilia's first $n$ : she is ahead with probability $a$ , behind with $a$ , tied with $1-2a$ . Her extra coin decides only the tie, giving $a+\tfrac12(1-2a)=\tfrac12$ .

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

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Lessons behind this question

Conditional Probability, Total Probability, and Bayes' Rule
Worked Examples — Probability Trees, Monty Hall, and Bayesian Updating
Sample Spaces, Events, and the Probability Axioms

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