Vishnu starts with $$1000$ and Jason with $$500$. They repeatedly toss a fair coin: on heads Vishnu pays Jason $$1$, on tails Jason pays Vishnu $$1$. They continue until one of them is broke. What is the probability that Jason is the one who goes broke?

Fair coin betting between 1000 and 500 dollars

Question

Vishnu starts with $\text{\char36}1000$ and Jason with $\text{\char36}500$ . They repeatedly toss a fair coin: on heads Vishnu pays Jason $\text{\char36}1$ , on tails Jason pays Vishnu $\text{\char36}1$ . They continue until one of them is broke. What is the probability that Jason is the one who goes broke?

Short answer

$\tfrac23$ . Jason's wealth is a fair game (a martingale), so its expected final value equals its starting value $500$ . At the end it is either $0$ or $1500$ , so $1500(1-p)=500$ and Jason is ruined with probability $p=\tfrac23$ .

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