To Flip Or Not To Flip

A fair coin, an unfair offer, and the price of certainty.

I sat down to work out a classic probability problem numerically, and accidentally built a casino.

The Problem of Points

In 1654, a gambler named Antoine Gombaud posed a question to Blaise Pascal: two players are in a race to win a certain number of points. The game is interrupted. How should they divide the pot?

Pascal wrote to Fermat, and their correspondence became one of the founding documents of probability theory. The answer is elegant: if you need a more points and your opponent needs b more, you can compute the fair split with a simple recurrence. Let P(a, b) be your probability of winning:

• P(0, b) = 1 — you just won
• P(a, 0) = 0 — your opponent just won
• P(a, b) = ½ · P(a−1, b) + ½ · P(a, b−1)

Every value in this table is a fraction with a power-of-2 denominator, and the numerators are just Pascal’s triangle. Beautiful math, clean solution, problem solved since the 17th century.

I built an interactive table to explore it. And then I thought: what if this were a game?

The Game

You and The House race to a target score. Each round, a fair coin is flipped — heads you score, tails The House scores. First to the target wins a pot of money.

But before each flip, judges look at the current game state, consult the probability table, and offer you cash to walk away. Accept, and you take the money. Decline, and the coin is flipped.

The question, every single round, is: to flip or not to flip?

You can play at willowdale.online/flip.

How the Judges Set Their Offers

The judges know the exact fair value of your position — they have the same formula Pascal and Fermat computed. If you have a 37.5% chance of winning a $10,000 pot, your fair value is $3,750.

But they don’t offer fair value. They offer the nearest “clean” fraction of the pot that sits strictly below your true odds.

“Clean” means small denominators whose only prime factors are 2, 3, and 5 — fractions like 1/3, 3/8, 7/20, nothing with a denominator above 20. These produce dollar amounts that look like something a human came up with: $3,333, $3,750, $3,500. Not $3,077 or $3,846, which look like someone ran the numbers to the last penny.

So if your fair value is $3,770 (193/512 of the pot), the judges offer $3,750 (3/8). Barely below fair, and a beautifully round number. If your fair value is $1,875 (3/16), they offer $1,666 (1/6). An 89% offer — a real discount, but still a clean, human-sounding number.

This matters psychologically. Round numbers feel like ballpark estimates — casual, generous, not fully analyzed. Precise numbers feel calculated. When the judges offer $7,500, it sounds reasonable. If they offered $7,517, you’d immediately suspect they did the math and it’s in their favor. The irony is that $7,517 is a better deal for you — but I think you’d be less likely to take it. The round number keeps your guard down.

The algorithm is deterministic — same game state, same offer every time. Just math dressed up in a game show contract.

Why People Sign

Since the offers are always strictly below fair value, the play that maximizes your expected winnings is to never accept a deal. The coin is fair, the game has zero house edge, and every offer leaves money on the table. A player who always flips would win 50% of their games and, on average, neither gain nor lose.

And yet.

When you’re ahead 4–3 in a race to 10, and the contract says $6,000, and you’ve already paid $5,000 to enter this game… you hesitate. That’s a guaranteed profit. The alternative is variance — maybe you win $10,000, but you are not that far ahead. Maybe your luck turns and you lose everything.

You know the offer is below fair. You can peek behind the curtain and see the exact numbers. The judges are shortchanging you by $128. But $128 feels like nothing when the alternative is watching your lead evaporate flip by flip.

So you sign. And $128 goes into the casino’s pocket.

This is what makes the game unusual. In blackjack or roulette, the house edge is baked into the rules — you can’t avoid it no matter how disciplined you are. Here, the game has no edge at all. The coin is fair. The race is symmetric. The only source of profit is human nature. Every dollar the casino makes is expected value that a player voluntarily left on the table.

Play for a while and you start to notice specific situations where the offer gets harder to refuse.

Managing risk. A guaranteed $7,500 is safer than a coin flip worth $7,734. In real life, you might need that money for rent. Variance has a real cost, and paying a premium for certainty can be entirely rational. There is a sophisticated argument for sometimes making decisions that reduce your expected value: bankroll management, survival probability and duration. Here the stakes are fictional, your bankroll buys nothing except more fair coin flips, and going broke is solved by refreshing your web browser, so that case is weaker — but it doesn’t feel weaker when your bankroll is shrinking and the judges are holding out real-looking money.

Mis-anchoring. The rational comparison is always between the offer and the expected value of continuing to flip. But that’s rarely the comparison your brain actually makes. If you were staring at a $0 offer last round and now the judges are offering $500, you’re comparing to the $0 — not to the $625 fair value. If your bankroll started at $10,000 and you’re down to $7,000, and the judges offer $3,200, you’re comparing to $10,000 — because taking the deal would put you above where you started. In both cases, the reference point that feels relevant has nothing to do with the expected value of this game.

Black and white thinking. When you’re behind in the race, the most likely single outcome is that you lose. If the judges offer $500 and your odds of winning are 6%, it feels like a choice between $500 and nothing. But expected value accounts for the 6% — the rare wins are big enough to compensate for all the losses across many games. You just don’t experience many games at once. You experience this one, where you’ll probably lose, and where the person who took $500 looks smart 94 times out of 100.

Imaginary momentum. You lose three flips in a row and it feels like the coin has turned against you — time to take the deal before things get worse. Or you win three in a row and feel like you’re on a streak that shouldn’t be interrupted. The coin has no memory. Each flip is independent. But the human brain is a pattern-recognition machine, and it will find narratives in random sequences whether they’re there or not.

The Optimal Judges

The judges in this game are clever, but simple — they mechanically pick the nearest clean fraction below fair value, blind to everything except the current expected value.

But the optimal offer would be very different. The right objective isn’t just the EV gap (fair value minus offer). It’s:

EV gap × P(acceptance | entire game trajectory)

A huge gap with low acceptance is worthless — the player just turns it down. A tiny gap with high acceptance is pennies. The sweet spot is a moderate discount the player almost can’t refuse.

And that acceptance probability depends on far more than just the current score — it depends on everything described above: the bankroll trajectory, the recent streak, what the last offer was, how long the player has been sitting there.

A perfect judge would think about all of this, and decide exactly what it can get you — tired, frustrated, scared little you — to accept. The clean-fraction heuristic doesn’t. And yet it still works. I still sign those offers.

The Lesson

The game is a playable demonstration of why casinos stay in business, maybe even why people accept below-market returns for safety, and why insurance companies are profitable.

The math is always available — right there behind a curtain. If your goal is to maximize expected dollars, the answer is always to flip the coin. And yet, round after round, the judges offer deals, and I sign them.

Play the game at willowdale.online/flip. It’s free, the coin is fair, and you will almost certainly take a deal you know you shouldn’t.