2. Probability and Fairness

Probability and Fairness

Probability provides a mathematical way to decide whether a game, competition, or random activity is fair.

In a simple two-player chance game, we often call the game fair when both players have the same probability of winning. More generally, when prizes, costs, or different outcomes are involved, fairness can be analysed using expected value: neither side should have a long-term mathematical advantage if the game is financially fair.

For example, suppose two players toss a fair coin.

• Player A wins if the coin lands heads.
• Player B wins if the coin lands tails.

Since:

P(heads) = 1/2

and:

P(tails) = 1/2

each player has the same probability of winning.

Therefore, the game is fair.

Probability and Fairness

For equally likely outcomes:

P(event) = number of favourable outcomes / total number of possible outcomes

A probability is always between:

0 and 1

where:

P(event) = 0 means impossible.

P(event) = 1 means certain.

For a fair six-sided die:

P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6

Equal probabilities of individual outcomes do not automatically mean that every game using the die is fair. The rules determine which outcomes cause each player to win.

A Simple Fair Dice Game

Suppose:

• Player A wins if an odd number is rolled.
• Player B wins if an even number is rolled.

The possible outcomes are:

{1, 2, 3, 4, 5, 6}

Player A wins with:

{1, 3, 5}

Therefore:

P(A wins) = 3/6 = 1/2

Player B wins with:

{2, 4, 6}

Therefore:

P(B wins) = 3/6 = 1/2

Both players have equal probabilities of winning.

Therefore:

the game is fair.

An Unfair Dice Game

Now change the rules:

• Player A wins by rolling 1, 2, 3, or 4.
• Player B wins by rolling 5 or 6.

Then:

P(A wins) = 4/6 = 2/3

and:

P(B wins) = 2/6 = 1/3

Player A is twice as likely to win.

Therefore:

the game is unfair.

Notice that the die itself is still fair. It is the rules of the game that create the unfairness.

Fair Random Devices

A random device is fair when its intended basic outcomes are equally likely.

Examples include:

• a balanced coin
• a properly constructed six-sided die
• a spinner divided into equal sections
• well-shuffled identical cards

For a spinner divided into four equal sections:

P(each section) = 1/4 = 25%

If the sections are unequal in size, their probabilities will generally not be equal.

That does not necessarily make a game unfair, however. The rules and rewards could compensate for the unequal probabilities.

Fairness Is About the Whole Game

Consider a spinner with:

• red probability = 1/2
• blue probability = 1/4
• green probability = 1/4

If:

• Player A wins on red
• Player B wins on blue or green

then:

P(A wins) = 1/2

and:

P(B wins) = 1/4 + 1/4 = 1/2

The spinner has unequal colour probabilities, but the game is fair between the two players.

This illustrates an important principle:

Fairness depends on the probabilities created by the complete rules, not simply on whether every individual outcome is equally likely.

Theoretical Probability

Theoretical probability is calculated from the mathematical structure of an experiment.

For a fair die:

P(rolling a 6) = 1/6

This probability does not mean that exactly one of every six rolls must be a 6.

It describes what we expect over many trials.

Experimental Probability

Experimental probability is calculated from actual results:

experimental probability = number of times event occurs / total number of trials

Suppose a die is rolled 60 times and 6 appears 13 times.

Then:

experimental P(6) = 13/60 ≈ 0.217

The theoretical probability is:

P(6) = 1/6 ≈ 0.167

These values are different.

That does not automatically mean the die is unfair.

Random variation is expected, especially with relatively small numbers of trials.

Short-Term Results Can Look Unfair

Suppose two people play a perfectly fair coin-toss game ten times.

Player A might win seven games.

Player B might win three.

That result does not prove that the game was unfair.

Fairness means that both players had equal probabilities before each game—not that they must win exactly the same number of times.

As the number of trials becomes very large, experimental proportions often tend to move closer to their theoretical probabilities. This is connected to the law of large numbers.

Expected Outcomes

Probability can be used to calculate how many times we expect an event to occur.

The basic relationship is:

expected frequency = probability × number of trials

Suppose a fair die is rolled 600 times.

The probability of rolling a 6 is:

1/6

Therefore:

expected number of 6s = 1/6 × 600

= 100

We expect approximately 100 sixes.

This does not mean there must be exactly 100.

Expected Outcome Example

A coin is tossed 200 times.

What is the expected number of heads?

P(heads) = 1/2

Therefore:

expected heads = 1/2 × 200

= 100

We would expect approximately 100 heads.

Results such as 96 heads or 107 heads would not be surprising.

Two-Stage Expected Outcomes

Suppose two fair coins are tossed.

The possible equally likely outcomes are:

HH, HT, TH, TT

There are four possibilities.

The probability of two heads is:

P(HH) = 1/4

If the experiment is repeated 200 times:

expected HH outcomes = 1/4 × 200

= 50

So we expect approximately 50 cases of two heads.

Expected Value

When different outcomes have different numerical rewards, we can calculate expected value.

For outcomes x₁, x₂, x₃, ... with probabilities P₁, P₂, P₃, ...:

E(X) = x₁P₁ + x₂P₂ + x₃P₃ + ...

In simpler words:

expected value = sum of (outcome value × probability)

Expected value describes the average result we would expect per play in the long run.

Expected Value Example

Suppose a game works like this:

• win $6 with probability 1/4
• win $2 with probability 1/2
• win $0 with probability 1/4

The expected winnings are:

E(X) = 6(1/4) + 2(1/2) + 0(1/4)

= 1.50 + 1.00

= $2.50

Therefore, the average expected payout is:

$2.50 per game

If the game costs exactly $2.50 to play, its expected net value to the player is:

$2.50 − $2.50 = $0

In the financial expected-value sense, the game is fair.

Expected Net Gain

For games involving a cost to play:

expected net gain = expected winnings − cost to play

If:

expected net gain = 0

the game is financially fair.

If:

expected net gain > 0

the game favours the player.

If:

expected net gain < 0

the game favours the organiser.

Example: Is the Game Fair?

A game costs $2 to play.

The possible prizes are:

Expected winnings:

E(X) = 10(0.05) + 3(0.20) + 0(0.75)

= 0.50 + 0.60

= $1.10

But the player pays:

$2.00

Therefore:

expected net gain = 1.10 − 2.00

= −$0.90

On average, the player would expect to lose 90 cents per play in the long run.

Therefore:

the game is not financially fair to the player.

Expected Value Does Not Predict One Game

Suppose the expected value of a game is:

−$0.90

This does not mean a player loses exactly 90 cents every time.

One player might win $10.

Another might win $3.

Another might win nothing.

Expected value describes the long-run average, not the result of an individual play.

This distinction is essential.

Comparing Two Games

Consider two games using a fair die.

Game A

Win $3 if you roll a 6.

Otherwise win $0.

Expected winnings:

E(A) = 3(1/6) + 0(5/6)

= $0.50

Game B

Win $1 if you roll an even number.

Otherwise win $0.

Probability of an even number:

3/6 = 1/2

Therefore:

E(B) = 1(1/2) + 0(1/2)

= $0.50

Although the games have very different rules, they have the same expected payout.

If both cost $0.50 to play, both are financially fair.

Probability Trees

When a game has several stages, a probability tree can help calculate the probabilities of combined outcomes.

For independent events, probabilities along a path are multiplied.

Suppose a fair coin is tossed twice.

Probability of HH:

P(HH) = 1/2 × 1/2 = 1/4

Probability of HT:

P(HT) = 1/2 × 1/2 = 1/4

Similarly:

P(TH) = 1/4

P(TT) = 1/4

Probability trees are especially useful when a game's fairness depends on combinations of several random events.

Fairness with Two Dice

Suppose two fair dice are rolled.

Player A wins if the total is 7.

Player B wins if the total is 8.

At first glance, the game may seem fair because each player is assigned one total.

But the totals are not equally likely.

There are 6 ways to make 7:

1 + 6
2 + 5
3 + 4
4 + 3
5 + 2
6 + 1

Therefore:

P(total 7) = 6/36 = 1/6

There are 5 ways to make 8:

2 + 6
3 + 5
4 + 4
5 + 3
6 + 2

Therefore:

P(total 8) = 5/36

Player A has the advantage.

The game is unfair.

Why Dice Totals Are Not Equally Likely

With two dice, there are:

6 × 6 = 36

equally likely ordered outcomes.

But different totals can be produced in different numbers of ways.

For example:

total 2 → 1 way

total 3 → 2 ways

total 4 → 3 ways

...

total 7 → 6 ways

...

total 12 → 1 way

Therefore, a game assigning one player a total of 7 and another a total of 12 would be extremely unfair.

Fairness with Spinners

Spinners are useful for designing probability games because probabilities can be changed by changing the sizes or numbers of sections.

For a spinner divided into eight equal sections:

If four sections are red:

P(red) = 4/8 = 1/2

If two are blue:

P(blue) = 2/8 = 1/4

If one is yellow:

P(yellow) = 1/8

If one is green:

P(green) = 1/8

The designer can use these probabilities to create either a fair or an unfair game.

Designing a Fair Spinner Game

Suppose an eight-section spinner contains:

• 4 red sections
• 2 blue sections
• 1 green section
• 1 yellow section

We could create this game:

Player A wins on red.

Player B wins on blue, green, or yellow.

Then:

P(A wins) = 4/8

= 1/2

and:

P(B wins) = (2 + 1 + 1)/8

= 4/8

= 1/2

Therefore, the game is fair.

The individual colours are not equally likely, but the players' total winning probabilities are equal.

Designing Fair Rewards for Unequal Probabilities

Fairness does not always require equal winning probabilities if the rewards are different.

Suppose:

• Event A occurs with probability 1/2 and pays $2.
• Event B occurs with probability 1/4 and pays $4.

Compare their expected payouts:

For A:

1/2 × $2 = $1

For B:

1/4 × $4 = $1

The less likely event has the larger reward.

This principle is often used when designing games with different prize levels.

Designing a Fair Game Using Expected Value

Suppose you want to design a game that costs:

$2 to play

A fair game should have:

expected winnings = $2

Suppose a spinner has four equally likely sections:

• $0
• $1
• $2
• unknown prize x

Then:

E(X) = 1/4(0) + 1/4(1) + 1/4(2) + 1/4(x)

For fairness:

E(X) = 2

So:

(0 + 1 + 2 + x)/4 = 2

3 + x = 8

x = 5

Therefore, the fourth prize should be:

$5

The expected payout becomes:

(0 + 1 + 2 + 5)/4

= 8/4

= $2

which matches the $2 cost.

Changing Rules to Make a Game Fair

Suppose a die game originally has:

• Player A wins on 1, 2, 3, or 4.
• Player B wins on 5 or 6.

The probabilities are:

P(A) = 4/6

P(B) = 2/6

One way to make the game fair is to change the rules:

• Player A wins on odd numbers.
• Player B wins on even numbers.

Now:

P(A) = 3/6 = 1/2

P(B) = 3/6 = 1/2

The game has become fair without changing the die.

Experimental Testing of Fairness

Suppose we design a spinner that should give:

P(A) = 1/2

P(B) = 1/2

We test it 20 times and obtain:

A wins 13 times.

B wins 7 times.

Experimental probabilities:

P(A) = 13/20 = 0.65

P(B) = 7/20 = 0.35

Does this prove the game is unfair?

No.

Twenty trials are not enough to expect perfect agreement with theoretical probability.

We should perform many more trials before concluding that the physical spinner itself might be biased.

Theoretical Fairness Versus Practical Fairness

A game can be mathematically fair in its rules but practically unfair if the random device is biased.

For example:

A coin game may have rules that give heads to Player A and tails to Player B.

Mathematically:

P(A) = P(B) = 1/2

But if the physical coin or tossing method strongly favours one outcome, the actual game may not behave as expected.

Therefore, evaluating fairness can involve both:

• theoretical probability
• experimental evidence

Randomness Does Not Mean Equal Results

A common misunderstanding is:

"If a game is fair, both players should win the same number of times."

That is not correct.

Fairness concerns the probability of winning, not a guarantee about actual results.

In a fair game, one player might win several times in a row simply because of random variation.

The Gambler's Fallacy

Suppose a fair coin lands heads five times in a row.

Someone might say:

"Tails is now more likely because it is due."

If each toss is independent, this is false.

The next toss still has:

P(heads) = 1/2

P(tails) = 1/2

Previous independent outcomes do not force future outcomes to "balance out" immediately.

This mistaken reasoning is called the gambler's fallacy.

Expected Value and Long-Term Advantage

Suppose a game has expected net value:

−$0.20 per play

One game may not seem important.

But after 1,000 plays:

expected total net result = 1,000(−$0.20)

= −$200

This explains why even a small probability advantage can become significant over many repetitions.

The expected value gives us a way to measure the game's long-term mathematical advantage.

Fair Does Not Mean Risk-Free

Consider two financially fair games.

Game A

• 50% chance of winning $1
• 50% chance of losing $1

Expected value:

E(A) = 0.5(1) + 0.5(−1)

= $0

Game B

• 50% chance of winning $100
• 50% chance of losing $100

Expected value:

E(B) = 0.5(100) + 0.5(−100)

= $0

Both games have expected value zero.

Both are mathematically fair in that sense.

But Game B involves much greater risk and variability.

Therefore, expected value tells us about long-term average fairness, not how risky a game feels to an individual player.

Evaluating a Game Systematically

When deciding whether a game is fair:

Identify all possible outcomes.

Make sure no outcomes are missing.

Determine the probability of each outcome.

Do not assume outcomes are equally likely unless there is a reason.

Identify which outcomes benefit each player.

Combine probabilities where necessary.

Include prizes and costs.

If different outcomes have different values, calculate expected value.

Compare the players' advantages.

For a simple win/lose game, compare winning probabilities.

For a prize game, compare expected net values.

Consider experimental evidence.

If a physical device is involved, repeated trials can help determine whether it behaves as expected.

State a justified conclusion.

Explain mathematically why the game is fair or unfair.

Example of a Complete Fairness Evaluation

A game costs $3.

A fair six-sided die is rolled.

• Roll 1: win $12
• Roll 2 or 3: win $3
• Roll 4, 5, or 6: win $0

Calculate expected winnings:

E(X) = 12(1/6) + 3(2/6) + 0(3/6)

= 2 + 1 + 0

= $3

The game costs:

$3

Therefore:

expected net gain = 3 − 3

= $0

So the game is financially fair in terms of expected value.

A player may still win or lose on an individual play, but neither the player nor organiser has a long-term expected monetary advantage.

Probability in Real Games

Probability and expected value are important when analysing:

• board games
• card games
• dice games
• carnival games
• prize draws
• raffles
• sports rules
• simulations
• insurance models
• business decisions

Carnival-style games are particularly useful examples because a game can appear attractive while its probabilities and payouts strongly favour the organiser.

The important mathematical question is not:

"Can I win?"

Instead, ask:

"What is the probability of each outcome, and what is its value?"

Common Mistakes

Assuming equal-looking choices have equal probabilities

Two labels on a spinner do not necessarily have the same probability if their sections have different sizes.

Assuming every dice total is equally likely

With two dice, a total of 7 is much more likely than a total of 2 or 12.

Judging fairness from one game

A fair game can still produce an uneven result in a small number of trials.

Confusing expected value with guaranteed value

An expected value of $2 does not mean the player receives $2 every time.

Ignoring the cost of playing

A $5 expected payout is not favourable if the game costs $8.

Ignoring outcomes with zero payoff

They still contribute to the probability distribution and must be included.

Assuming a rare outcome automatically makes a game unfair

A rare outcome can have a larger reward that compensates for its lower probability.

Using experimental probability from too few trials

Small samples can differ considerably from theoretical probability because of random variation.

Falling for the gambler's fallacy

Independent events do not become more or less likely simply because of previous outcomes.

Key Terms

Probability — a numerical measure of how likely an event is to occur.

Outcome — one possible result of a probability experiment.

Sample space — the complete set of possible outcomes.

Event — one or more outcomes of interest.

Fair game — a game in which the rules do not give one participant an unintended mathematical advantage; in monetary games, fairness is commonly assessed by expected net value.

Unfair game — a game whose probabilities, rules, or rewards give one side a mathematical advantage.

Theoretical probability — probability calculated from a mathematical model.

Experimental probability — probability estimated from observed results.

Expected frequency — the predicted number of occurrences of an event over a specified number of trials.

Expected value — the probability-weighted average numerical outcome over many repetitions.

Expected net gain — expected winnings minus the cost of participating.

Independent events — events for which the occurrence of one does not change the probability of the other.

Probability tree — a branching diagram showing possible sequences of events and their probabilities.

Random variation — natural differences between actual results and theoretical expectations.

Law of large numbers — the principle that experimental proportions tend to approach theoretical probabilities as the number of independent trials becomes very large under suitable conditions.

Gambler's fallacy — the mistaken belief that previous independent random outcomes change what is "due" next.

Key Takeaways

• Probability can be used to determine whether games and activities are mathematically fair.
• In a simple two-player chance game with equal rewards, equal probabilities of winning generally produce a fair game.
• A fair random device does not automatically create a fair game; the rules also matter.
• Unequal individual outcome probabilities can still be combined to produce a fair game.
• Expected frequency is calculated using probability × number of trials.
• Expected value is calculated by multiplying each outcome's value by its probability and adding the results.
• A financially fair game has an expected net gain of 0.
• A rare event can have a larger reward and still form part of a fair game.
• Expected value describes a long-run average, not what must happen in one play.
• Experimental results will not usually match theoretical probabilities exactly.
• Short runs of wins or losses do not necessarily indicate an unfair game.
• With repeated trials, experimental proportions tend to become more stable.
• Games can be made fair by adjusting probabilities, rules, costs, or rewards.
• A strong evaluation of fairness considers all possible outcomes, their probabilities, their rewards or costs, and the assumptions behind the probability model.

Check Your Understanding

1. A fair coin is tossed. Player A wins on heads and Player B wins on tails. Explain why the game is fair.
2. A fair die is rolled. Player A wins on 1, 2, 3, or 4. Player B wins on 5 or 6. Calculate each player's probability of winning and determine whether the game is fair.
3. Design new rules for the die in Question 2 that give both players the same probability of winning.
4. A spinner has eight equal sections: four red, two blue, one yellow, and one green. Player A wins on red. Player B wins on any other colour. Determine whether the game is fair.
5. A fair die is rolled 300 times. Calculate the expected number of times a 5 should appear.
6. Two fair coins are tossed 400 times. Calculate the expected number of times two heads should occur.
7. A game has a 1/4 probability of paying $8 and a 3/4 probability of paying $0. Calculate the expected payout.
8. The game in Question 7 costs $2 to play. Calculate the expected net gain and determine whether the game is financially fair.
9. A game costs $3. A player has a 1/2 probability of winning $4, a 1/4 probability of winning $2, and a 1/4 probability of winning nothing. Calculate the expected winnings and determine whether the game is fair.
10. A game uses two dice. Player A wins if the total is 7 and Player B wins if the total is 12. Explain why assigning one total to each player does not make the game fair.
11. A theoretically fair spinner is tested 20 times. Player A wins 14 times and Player B wins 6 times. Explain why this result alone does not prove that the spinner is unfair.
12. A fair coin has landed heads six consecutive times. A student claims that tails must now be more likely. Evaluate the student's reasoning.
13. A four-section spinner has equally likely prizes of $0, $2, $4, and x dollars. The game costs $3 to play. Determine x so that the game is financially fair.
14. Design your own fair game using a six-sided die. State the rules and use probabilities or expected value to prove mathematically that your game is fair.
15. A carnival game advertises a large jackpot and claims, "Everyone has a chance to win, so the game is fair." Explain why having some chance of winning is not enough to establish mathematical fairness.