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#53 Coin-flip traders: can luck beat the fees? No edge after fees
Past prices; no real orders were placed. Every figure is a share of the money, or a count.
Last checked: 18 October 2026.
Where the idea comes from
- The law of large numbers. Jacob Bernoulli (1713) proved that the share of heads in many fair flips closes in on one half.
- How far luck wanders. Abraham de Moivre (1738) showed that the running count of heads less tails spreads out like the square root of the number of flips: the bell curve.
- Prices as a coin. Louis Bachelier (1900) first modelled prices as a random walk. Burton Malkiel's A Random Walk Down Wall Street (1973) made famous the idea that a blindfolded monkey throwing darts would pick stocks about as well as the experts.
- Why every test needs a random twin is on our theory page "Hard-to-beat markets, and why every test needs a random twin". This page builds that twin for trading, and opens the series "Published edges against the coin": each later study takes a published finding and asks whether it beats these traders.
The question
A trader decides by coin flips. One flip (or three) says whether to trade today, and one more says long or short. There are three of them, one each in the Asian, London and New York sessions. Over day after day and year after year, does the law of large numbers even it out? And can such a trader beat the fees?
How we tested
- Three coins: Bitcoin, Ether and Solana. Hourly futures prices (USDT perpetuals), 15 October 2022 to 15 October 2026.
- Three sessions, in UTC: Asia 00:00 to 08:00, London 08:00 to 16:00, New York 13:00 to 21:00.
- Two ways to trade:
- Two trades a day: the session cut into two 4-hour trades, each one at 1x (no leverage). One flip to trade or skip, one for the side.
- One trade a day, held all session: 3x leverage, all of the money, no stop. Three gates were tried before the side flip:
- one flip: trade on heads;
- best of 3: trade on two heads or more;
- all 3 heads: trade only on three heads. A trade that moves 32.8% against it inside the session is liquidated and the trader is out.
- Fees: the exchange's market-order fee (0.055%) and its resting-order fee (0.02%), paid on the way in and on the way out. A no-fee run shows pure luck.
- 10,000 traders for every coin, session and way to trade, each with their own flips: 90,000 for each row of the chart below.
- The pure coin: no market at all. Win 1 or lose 1 on each trade, two chances a day, for 40 years, 10,000 traders.
- Nothing was tuned, so there was nothing to hold back. Each half of the four years was also run on its own, as a check.
The maths, step by step
1. A coin-flip side earns nothing. If the price moves 2%, the long gains 2% and the short loses 2%. With the side chosen by a coin, the average is 0%.
2. The fee is certain. In and out costs 0.055% twice, 0.11% of the trade's size. At 3x the size is three times the money, so 0.33% of the money every trade.
3. Fees, many times over. 1,459 trades that each lose 0.11% leave 0.9989 multiplied by itself 1,459 times, about 20% of the money: a loss of 80%. At 3x, 730 trades of 0.33% leave about 9%: a loss of 91%.
4. The swings cost too. Down 10% then up 10% is not even: 0.90 × 1.10 = 0.99. At 3x a 3% move becomes 9%, and 0.91 × 1.09 = 0.992. So even with no fee, the middle trader slowly loses.
5. What the law of large numbers says. With trades of win 1 or lose 1, the running total after n trades spreads out like the square root of n. 95% of traders end within about twice that of zero.
- After a year: about 365 trades, square root about 19, so within about ±38.
- After 40 years: about 14,600 trades, square root about 121, so within about ±240.
The average per trade is the total divided by the trades: ±0.10 after a year, ±0.017 after 40. The average evens out. The total doesn't.
6. Three flips. Of the 8 ways three flips can land, 4 have two heads or more (HHH, HHT, HTH, THH). So best of 3 trades 1 day in 2, the same as one flip. Only HHH is three heads: 1 day in 8.
What we found
1. At the market-order fee, about 1 trader in 100 ended ahead.
- Two trades a day, 1x: 0.3% of 90,000 ended ahead. The middle trader lost 83%, close to step 3's 80% from fees alone. No Bitcoin trader of the 30,000 ended ahead.
- All session, 3x, one flip: 1.0% ended ahead. The middle trader lost 97.5%.
- Each half on its own gave the same picture: 2.2% and 1.4% ahead for two trades a day.
2. Three flips only helped by trading less.
- Best of 3 traded as often as one flip, and ended the same: 1.0% ahead.
- All 3 heads traded about 180 times instead of 730. That's a quarter of the fees, so 11.7% ended ahead and the middle trader lost 62%. Fewer trades meant a smaller loss, not a profit.
3. Even with no fee at all, fewer than half ended ahead.
| market-order fee | resting-order fee | no fee | |
|---|---|---|---|
| two trades a day, 1x | 0.3% | 8.5% | 39.0% |
| all session, 3x, one flip | 1.0% | 7.9% | 20.6% |
| all session, 3x, best of 3 | 1.0% | 7.8% | 20.4% |
| all session, 3x, all 3 heads | 11.7% | 24.4% | 34.2% |
Share of 90,000 traders ending ahead.
At the market-order fee about 1 trader in 100 ends ahead (1 in 9 of those who trade one day in eight), and even with no fee at all fewer than half do.
4. At the fee the session didn't matter; the coin did. London, Asia and New York gave about the same results. The wilder the coin, the wider the spread: the luckiest 1% of Solana's two-trades-a-day traders in New York were up 2.2%, Bitcoin's luckiest 1% were still down 46% or more.
5. 3x was safe on Bitcoin and Ether, and not on Solana.
- No Bitcoin or Ether trader was liquidated: their biggest move inside a session was 16% and 27.5%.
- On Solana, 4,389 of 10,000 one-flip Asia traders were liquidated: on 10 November 2022, in the FTX collapse (2,519), and on 14 January 2023 (1,870). In New York, 4,400: on 8 and 9 November 2022.
- In London the biggest move, 32.8% on 9 November 2022, stopped just short of the liquidation line.
6. The law of large numbers, as promised, and no more. In the pure coin game the middle trader stayed at 0, while the luckiest and unluckiest 2.5% drifted further apart every year: ±38 after a year, about ±119 after ten, about ±238 after forty, as step 5 says. The average per trade closed in on 0: ±0.10, then ±0.032, then ±0.016.
The middle stays at 0 while the luckiest and unluckiest drift further apart every year.
The average per trade closes in on 0 year by year, which is all the law of large numbers promises.
What it means
- A coin can't beat the fee. On a coin-flip side every trade's average is minus the fee, and over hundreds of trades that is close to certain.
- The law of large numbers evens out the average, not the account. A coin-flipper can be far ahead or far behind for years by luck alone.
- The only way past the fee is to know something: a reason for the side that beats 50/50 by more than the costs, on prices the rule hasn't seen. That is the bar for the rest of this series.
Limits
- Three large coins, four years, hourly prices: a liquidation is judged on the hour's high and low, not on every tick.
- Funding is left out. A session-long trade pays it at most once, and on a coin-flip side it averages out.
- The fees are the exchange's base rates. Larger traders pay less, but the no-fee column shows that even zero isn't enough.
Credits
- Jacob Bernoulli, Ars Conjectandi, 1713: the law of large numbers.
- Abraham de Moivre, The Doctrine of Chances, second edition, 1738: the square-root spread of the running count, and the bell curve.
- Louis Bachelier, Théorie de la spéculation, 1900: prices as a random walk.
- Burton G. Malkiel, A Random Walk Down Wall Street, 1973: the dart-throwing monkey.
- The coin-flip traders, the three sessions and the three-flip gates are our own design.
- Data: Bybit's public market data, 1-hour USDT perpetual prices; fees from Bybit's base fee schedule.
- PerpGremlin and Claude, Coin-flip traders: can luck beat the fees?, TerminalGremlin, 2026: the tests, the code and the words.
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