Variance

Lesson 8 of 12

Variance is one of the most important concepts in probability and gambling mathematics.

It explains why actual results can differ significantly from expected results, especially over short periods of time.

In roulette, variance is responsible for both winning streaks and losing streaks.

What Is Variance?

Variance measures how far actual outcomes can deviate from the expected outcome.

Even when the Expected Value is known, individual results may vary considerably.

For example:

Expected outcome after 100 spins:

  • Loss of €2.70 for every €100 wagered

Actual outcome:

  • Profit of €150

  • Loss of €300

  • Break-even

All of these results are possible because of variance.

Why Variance Exists

Roulette is a random process.

Every spin produces uncertainty.

Although probability determines long-term expectations, short-term results remain unpredictable.

This uncertainty creates fluctuations around the expected result.

These fluctuations are called variance.

Winning Streaks and Losing Streaks

Variance creates sequences that appear unusual.

Examples:

  • Red appears 10 times in a row

  • Black appears 12 times in a row

  • A player wins 15 spins out of 20

Such events occur naturally in random systems.

They do not indicate a broken wheel or a winning strategy.

Variance and Betting Systems

Many betting systems appear successful because of variance.

A player may use:

  • Martingale

  • Fibonacci

  • D'Alembert

  • Labouchere

and experience a profitable session.

This does not necessarily mean the system has an advantage.

Variance can create positive results even when Expected Value remains negative.

Small Samples vs Large Samples

Variance has a greater impact on small samples.

Example:

10 Spins:
Results may differ dramatically from expectation.

100 Spins:
Variance remains significant.

10,000 Spins:
Expected Value becomes more dominant.

As sample size increases, the influence of variance becomes relatively smaller.

Casino Perspective

Casinos understand variance very well.

In the short term:

  • Players can win

  • Casinos can lose

In the long term:

  • Expected Value becomes dominant

  • Results move closer to mathematical expectations

This is why casinos focus on volume rather than individual sessions.

Variance vs Expected Value

Expected Value:

  • Long-term average outcome

Variance:

  • Short-term fluctuations around that average

Both concepts are necessary to understand roulette correctly.

Expected Value explains where results tend to go.

Variance explains how they get there.

Why This Matters

Without understanding variance, players often mistake luck for skill.

A profitable week may simply be the result of random fluctuation.

Likewise, a losing week does not necessarily mean a strategy is flawed.

Understanding variance helps players evaluate results more objectively.

Key Facts

  • Variance creates short-term fluctuations

  • Winning streaks and losing streaks are normal

  • Variance can temporarily hide Expected Value

  • Small samples are heavily influenced by variance

  • Large samples move closer to expectation

Next Step

Continue with Standard Deviation to learn how statisticians measure the size of variance and uncertainty.

Previous: Expected Value

Next: Standard Deviation