The Law of Large Numbers — What 2,000 Simulations Reveal About Going Broke

Chapter 2 calculated expected value. But expected value is an average. Even with a negative average, couldn’t a lucky player still come out ahead?

This chapter answers that question with a simulation — not a calculation, but an actual run of the numbers.

The setup

We built a program that repeats baccarat banker bets under these conditions:

• Starting bankroll: $1,000
• Bet per hand: $10
• Win probability 45.86%, with a 5% commission on wins
• If the bankroll drops below the bet size, the run is marked as bust and stopped
• Each condition simulated 2,000 independent players

In other words, this measures: “If 2,000 people start with the same money and play the same game, how many survive?”

Results

Hands played	Bust rate	Average remaining balance
100	0.0%	$894.82
1,000	62.1%	$98.67
10,000	100.0%	$0.00

The most important row in this table

The 100-hand row. 0% bust rate.

Nobody who played 100 hands went broke. On average they lost about $10, but plenty of them were ahead. At this stage, gambling looks like a game worth playing.

And that’s exactly the trap. What happens in your first 100 hands tells you nothing about what’s coming — if anything, it builds the opposite conviction.

What is the Law of Large Numbers?

The Law of Large Numbers says that as the number of trials increases, actual results converge toward the expected value.

• Flip a coin 10 times and you might get 7 heads (70%).
• Flip it 10,000 times and the share of heads will almost certainly land near 50%.

Applied to gambling, this means:

Luck doesn’t disappear. But as more hands are played, luck’s relative influence shrinks, leaving only the steady 1.06% leaking out of every hand.

The bankroll mismatch — where the real contest is decided

There’s a second structural piece layered on top of this: the operator and the individual player don’t have the same amount of money.

• The operator’s bankroll is effectively unlimited.
• An individual’s bankroll is finite.

Even in a perfectly fair game with an expected value of exactly zero, the side with less money hits zero first. This is known as the Gambler’s Ruin problem.

”Then just quit when you’re ahead”

That’s true in theory. In practice, it almost never happens, for two reasons.

First, there’s no way to know when you’ve peaked. Quitting while ahead requires knowing “this is the top” — and that information doesn’t exist.

Second, winning doesn’t make people quit — it makes them keep going. This comes down to the brain’s reward wiring, covered in Chapter 6. People who have actually won tend to bet longer and bigger, not less.

Look again at the 1,000-hand average balance of $98.67. That’s an average loss of about 90%. Every one of those players had moments of being ahead along the way.

Summary

1. At 100 hands, nobody goes broke. That’s why people start.
2. At 1,000 hands, 62% go broke. By this point, it’s already hard to stop.
3. At 10,000 hands, 100% go broke. No exceptions in this simulation.
4. The side with a finite bankroll goes broke eventually, even in a perfectly fair game.

Next chapter, we look at the attempt to beat this conclusion — betting strategies — and why they actually speed up bankruptcy rather than prevent it.

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Sources

• Figures in this chapter come from a seeded Monte Carlo simulation (2,000 independent trials per condition).
• Win rate and commission follow standard baccarat rules; actual results vary with house rules and betting style.
• Verified: 2026-07-29

If gambling is a problem for you — in Korea, call the Gambling Problem Helpline at 1336, toll-free, 24/7, free of charge. If you’re outside Korea, search for your country’s national problem-gambling helpline.