The Infinite Startup Payout
A probability puzzle about expected value, real-world constraints, and why math without context lies politely.
A game has infinite expected value on paper. You still would not pay infinite money to play it.
Puzzle
Try it before the answer.
A fair coin is tossed until the first heads. If heads appears on toss n, you win 2^n rupees. The expected value is mathematically infinite. Should you pay any finite amount to play?
Hints
- The expected value calculation is not the whole decision.
- Ask who can actually pay the largest outcomes.
- Utility is not linear in money.
Solution
The formal expected value diverges, but real decisions have bankroll limits, payout limits, risk preferences, and diminishing utility. The puzzle separates algebra from decision-making.
Algebra Meets Reality
The payout grows quickly, but the person offering the game has a bankroll and may have a payout cap. Compare those limits with the formal expected value before deciding what to pay.
Sources & further reading
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