A Game is Said to Be Fair If (It’s Not What You Think)

We have all heard the phrase, “That’s not fair!”

In casual conversation, a “fair game” means everyone has an equal shot at winning. But in mathematics, statistics, and game design, the definition is much more precise—and often surprising.

You might assume that flipping a coin is fair because it has a 50/50 outcome. But what if you win $1forheadsbutlose$2 for tails? Is that still fair?

In this post, we will break down the exact mathematical condition for fairness. By the end, you will understand the critical concept of Expected Value and why most “fair” games in casinos are actually designed to be unfair.

A Game is Said to Be Fair If… (The Core Definition)

In probability theory and gambling mathematics, a game is said to be fair if the expected value (EV) of the game is zero.

Let’s unpack that.

• If EV > 0: The game favors the player. (You should play)

• If EV = 0: The game is perfectly fair.

• If EV < 0: The game favors the house. (The casino wins over time)

Decoding “Expected Value”

Expected Value is the average outcome you would expect if you played a game an infinite number of times. It accounts for three things:

1. The probability of winning

2. The probability of losing

3. The payoff (winnings) versus the cost (losses)

The Fairness Formula:

E = (Probability of Win × Amount Won) + (Probability of Loss × Amount Lost)

If E = 0, the game is mathematically fair.

Fair Game Example #1: The Honest Coin Flip

Let’s test the definition.

You bet $1 on a coin flip.

• Heads (Win): You keep your $1andwinanadditional$1 (Net profit = +$1)

• Tails (Loss): You lose your $1(Netprofit=-$1)

The Math:

• Probability of Win = 0.5

• Probability of Loss = 0.5

• EV = (0.5 × $1)+(0.5\times -$1)

• EV = ($0.50)+(-$0.50)

• EV = $0.00

Conclusion: This coin flip game is perfectly fair. Over 1,000 flips, you will likely break even (ignoring variance).

Unfair Game Example: The Casino Roulette Trap

Now, let’s see why the house always wins. Consider European Roulette (single zero).

You bet $1 on Red. There are 18 red slots, 18 black slots, and 1 green slot (zero). Total possible outcomes = 37.

• Win (Red): You win $1(Profit=+$1)

• Loss (Black or Green): You lose $1(Profit=-$1)

The Math:

• Probability of Win = 18/37 ≈ 0.4865

• Probability of Loss = 19/37 ≈ 0.5135

• EV = (0.4865 × $1)+(0.5135\times -$1)

• EV = $0.4865-$0.5135

• EV = -$0.027 (Negative 2.7 cents)

Conclusion: Every time you bet $1 on Roulette red, you expect to lose 2.7 cents. This game is unfair (specifically, it has a “negative expectation”).

Why “Fair” Doesn’t Mean “50/50”

This is the biggest misconception. A game can have wildly different probabilities but still be mathematically fair.

Example: A dice game where you win $6ifyourolla6,butlose$1 if you roll 1-5.

• Probability of Win (Roll 6) = 1/6 ≈ 16.6%

• Probability of Loss (Roll 1-5) = 5/6 ≈ 83.4%

• EV = (1/6 × $6)+(5/6\times -$1) = $1+(-$0.83) = $0.00

This is fair! Even though you lose most of the time, the rare win pays enough to balance the risk.

Key Takeaway: Fairness is about risk versus reward, not frequency of winning.

Beyond Mathematics: Regulatory Fairness

While a mathematician focuses on the Expected Value of $0, regulators and lawyers define “a game is said to be fair if” it meets three legal criteria:

1. Randomness (RNG Certification)

The outcome cannot be predicted or manipulated. In video games and online casinos, a certified Random Number Generator (RNG) must be used.

2. No Hidden Information

All players must know the rules and the odds before they play. A game where one player knows the future (e.g., a marked deck) is definitionally unfair.

3. Consistency

The rules cannot change mid-game. You cannot decide after a loss that “we are playing best 2 out of 3 now.”

The “Fair Game” Fallacy in Gambling

Here is the hard truth for gamblers: No commercial casino game is mathematically fair.

Every slot machine, roulette wheel, blackjack shoe, and craps table has a built-in House Edge (negative expected value for the player). Casinos stay in business because they statistically never offer a game where EV = 0.

The only technically “fair” games in a casino are:

• Poker (You play against other players, not the house. The casino takes a “rake” – but the game between players can be fair if skill is equal).

• Perfect Blackjack Basic Strategy (even this only reduces the edge to ~0.5% – still technically unfair).

How to Spot a Fair Game (Checklist for Players)

Before you put your money or time into any game, run this checklist:

• Calculate the EV: Is the mathematical return actually zero? (Be honest with probabilities).

• Check for Rake/Fees: Even if EV = 0, transaction fees, entry fees, or “the rake” make the game unfair.

• Verify the RNG: Is the shuffle truly random? Is the dice roll physically verifiable?

• No Insider Knowledge: Is every player seeing the same board/hand/cards?

If all four boxes are checked, you have found a perfectly fair game.

Conclusion: Play Smart, Not Just “Fair”

To summarize definitively:

A game is said to be fair if the expected value of the outcome equals zero, meaning that over infinite plays, neither the player nor the opponent has a mathematical advantage.

However, remember that mathematically fair games are rare in the real world. Casinos and game developers need to make a profit, so they build in a negative EV. Video game loot boxes often hide their negative EV behind “random” rewards.

Your best strategy? Never play a game expecting to beat the math. If the EV is negative, treat the activity as entertainment with a cost. If the EV is zero, enjoy the pure risk, but know that luck—not skill—will decide the winner.