Worked solutionOne step at a time
The order of flips does not matter
Every flip multiplies your wealth. Multiplication does not care about order, so after W wins and L losses your wealth is the same however they were arranged.
A typical player wins half the flips
On a fair coin the typical (median) player has about as many wins as losses. Group them in pairs of one win and one loss.
Turn the pair into a per-flip rate
A pair is two flips, so the per-flip factor is the square root. This is the geometric mean, the growth rate one player actually experiences.
Answer: C, about −5% a flip
The arithmetic mean of the two outcomes, 1.05, is the growth of the average across many players. The geometric mean, 0.949, is the growth of a typical single player over time. After 100 flips the typical player has $100 × 0.949100 ≈ $0.52.
What the simulation showed10,000 players, 100 flips
The reel flips 100 fair coins for each of 10,000 players who start with $100 (fixed seed, every number on screen read from the run).
- 87.1% of players finish below $100.
- The average wealth is $6,159, because the richest 1% hold 97% of all the money. In theory the average is $100 × 1.05100 ≈ $13,150; a sample of 10,000 rarely contains the handful of extreme winners that carry it, so the sample average comes out lower.
- Break-even needs 56 wins out of 100: 55 wins leave you with about $50, 56 wins with $126. Only 12.9% of players got there.
- The median player ended with $0.52, which is −5.1% per flip, the same as √0.9.
Wrong answersWhy each one is tempting
| Pick | What went wrong |
|---|---|
| A · grows 5% | Used the average over players (the ensemble average) instead of the growth one player sees over time. |
| B · stays flat | Assumed a 50% gain and a 40% loss roughly cancel. They give 0.9, a 10% loss per pair. |
| D · shrinks 10% | Correct factor, wrong time scale: 0.9 is per two flips. |
Follow-ups an interviewer asks nextTry these before opening them
You may bet only part of your wealth each flip. What fraction grows fastest?
Bet a fraction f: the outcomes are 1 + 0.5f and 1 − 0.4f. Maximise ½ ln(1 + 0.5f) + ½ ln(1 − 0.4f). The derivative is zero at f = 0.25, so bet 25% each flip. The typical growth is then about +0.6% per flip, positive.
Ten players pool their money and split it evenly after every flip. Does that help?
Yes. Pooling averages away part of the luck, so the group's growth moves from the geometric mean towards the arithmetic mean. With enough players sharing, each one grows close to the +5% average.
What is the probability a single player is up after 100 flips?
They need at least 56 wins out of 100. For a fair coin that is about 13.6%, which matches the 12.9% the simulation measured.
Where else does arithmetic vs geometric mean bite?
Anywhere returns compound: a +50% year followed by a −50% year leaves you 25% down, although the average return is 0%. Volatility drags compound growth by roughly σ²/2 per period.
