2. Games of Chance

Games of Chance

Introduction to Games of Chance

Games of chance are activities in which outcomes depend partly or entirely on random events. Examples include lotteries, dice games, card games, roulette, and prize draws.

Although individual outcomes are uncertain, mathematics allows us to analyse how these games work.

Probability can help answer important questions:

• What is the probability of winning?

• How much should a fair game pay?

• What is the expected profit or loss from playing?

• How do gambling operators make money?

• Can a betting strategy change the expected outcome?

• What happens when a game is repeated many times?

The key mathematical concepts are probability, expected value, fair payouts, house advantage, and long-term outcomes.

Understanding these concepts allows us to evaluate games of chance objectively and distinguish between short-term luck and long-term mathematical expectations.

Understanding Games of Chance

A game of chance is a game in which randomness affects the result.

Some games depend almost entirely on chance, while others combine chance with skill.

Examples include:

To analyse a game mathematically, we must identify:

• The possible outcomes.

• The probability of each outcome.

• The cost of playing.

• The payout associated with each outcome.

• The player's net gain or loss.

Gross Payout Versus Net Profit

These two quantities must be distinguished.

Gross payout is the amount received when a prize is awarded.

Net profit is the payout minus the cost of playing.

For example, if a ticket costs $5 and pays $20:

Net profit = $20 − $5

Net profit = $15

If the ticket does not win:

Net profit = $0 − $5

Net profit = −$5

This distinction is essential when calculating expected value.

Expected Value in Games of Chance

Expected value measures the probability-weighted average of possible outcomes.

For a discrete random variable X:

E(X) = Σ[x × P(X = x)]

where:

• E(X) is the expected value.

• x is a possible outcome.

• P(X = x) is the probability of that outcome.

• Σ means to add all the probability-weighted outcomes.

For games of chance, X often represents the player's net profit.

Interpretation

If a game has an expected value of −$2 per play, the player is expected to lose an average of $2 per play over a very large number of repetitions.

This does not mean the player loses exactly $2 every time.

Some plays may produce large wins, while others produce losses.

Expected value describes the long-term mathematical average.

Worked Example: A Simple Dice Game

A player pays $3 to roll a fair six-sided die.

If the die shows a 6, the player receives $12.

Otherwise, the player receives nothing.

Step 1: Identify the Probabilities

P(rolling a 6) = 1/6

P(not rolling a 6) = 5/6

Step 2: Determine Net Outcomes

If the player wins:

Net profit = $12 − $3 = $9

If the player loses:

Net profit = −$3

Step 3: Calculate Expected Value

E(X) = (1/6 × $9) + (5/6 × −$3)

= $1.50 − $2.50

E(X) = −$1

Interpretation

The player loses an average of $1 per game in the long run.

The game favours the operator.

Fair Games

A fair game is a game in which the expected net profit is zero.

E(X) = 0

In a fair game, neither the player nor the operator has a mathematical advantage in expected monetary value.

Example: Fair Coin-Toss Game

A player pays $5 to play.

A fair coin is tossed.

If heads appears, the player receives $10.

If tails appears, the player receives nothing.

The possible net outcomes are:

Expected value:

E(X) = 0.5 × $5 + 0.5 × (−$5)

= $2.50 − $2.50

E(X) = $0

The game is fair in terms of expected monetary value.

Important Distinction

A fair game does not guarantee that each player breaks even.

Individual players can still win or lose.

Fairness refers to the mathematical expectation over repeated plays.

Calculating Fair Payouts

A fair payout is the prize amount that makes the expected net profit equal to zero.

Suppose a game costs $4 to play.

The probability of winning is 1/8.

The player receives a gross payout of W dollars if successful and nothing otherwise.

Step 1: Construct the Expected Value Equation

E(X) = (1/8)(W − 4) + (7/8)(−4)

For a fair game:

E(X) = 0

Step 2: Solve

0 = (W − 4)/8 − 28/8

0 = (W − 32)/8

W = $32

Interpretation

The fair gross payout is $32.

The player's net profit when winning would be:

$32 − $4 = $28

The probability of winning is 1/8, so the expected gross payout is:

1/8 × $32 = $4

This exactly equals the cost of playing.

General Formula for a Fair Payout

Suppose:

• The game costs C dollars.

• The probability of winning is p.

• The winner receives a gross payout W.

• All other outcomes pay nothing.

The expected net profit is:

E(X) = pW − C

For a fair game:

0 = pW − C

Therefore:

W = C/p

Example

A lottery ticket costs $2.

The probability of winning is 1/500.

The fair gross payout is:

W = 2/(1/500)

W = $1,000

A gross prize of $1,000 would make this simplified lottery fair, assuming there are no other prizes or costs.

Fair Odds

Fair odds describe the relationship between the probability of winning and the payout required for a fair game.

Suppose an event has probability p.

The fair gross payout multiplier per unit staked is:

1/p

The fair net profit multiplier on a winning bet is:

(1 − p)/p

Example

An event has probability:

p = 0.20

Fair gross payout multiplier:

1/0.20 = 5

For a $10 stake:

Fair gross payout = $50

Fair net profit if successful:

$50 − $10 = $40

The fair net odds are 4 to 1.

Important Distinction

A quoted payout may refer to the total amount returned or only the profit.

Always determine whether the original stake is included.

Lotteries and Combinations

Lotteries often involve selecting several numbers from a larger set.

Combinations are useful when the order of selection does not matter.

The number of ways to select r objects from n objects is:

C(n,r) = n!/[r!(n − r)!]

where ! represents factorial.

Worked Example: A Six-Number Lottery

Suppose a lottery requires selecting six numbers from 49.

The order of the selected numbers does not matter.

The number of possible combinations is:

C(49,6) = 49!/(6!43!)

C(49,6) = 13,983,816

If one combination wins the jackpot:

P(jackpot) = 1/13,983,816

This is approximately:

0.00000715%

The probability of winning is extremely small.

Calculating a Fair Lottery Jackpot

Suppose the six-number lottery costs $2 per ticket.

There is only one prize: the jackpot.

The probability of winning is:

p = 1/13,983,816

The fair gross payout is:

W = C/p

W = 2 × 13,983,816

W = $27,967,632

Under this simplified model, a jackpot of approximately $27.97 million would make the expected net value of a ticket zero.

Important Limitations

Real lotteries often include:

• Multiple prize categories.

• Shared jackpots.

• Taxes.

• Administrative costs.

• Different ticket prices.

• Jackpot rollover rules.

These factors affect the expected value.

A jackpot larger than the simple fair amount does not necessarily make a real lottery favourable, especially when the prize may be shared among several winners.

House Advantage

The house advantage, also called the house edge, measures the operator's expected profit as a percentage of the player's stake.

For a game with stake C:

House edge = −E(player net profit)/C × 100%

This assumes the game is analysed from the player's perspective.

Example

Suppose a game costs $10 to play.

The expected net profit for the player is:

E(X) = −$0.80

Then:

House edge = −(−0.80)/10 × 100%

House edge = 8%

Interpretation

The operator expects to earn an average of 8% of the amount wagered over many plays.

This does not mean the operator earns exactly 8% from every player or every game.

Actual results vary because of randomness.

Roulette and House Advantage

Roulette provides a useful example of how the house edge is created.

In European roulette, there are 37 numbered pockets:

0 through 36.

Suppose a player places a $1 bet on one specific number.

The probability of winning is:

P(win) = 1/37

The probability of losing is:

P(loss) = 36/37

A winning straight-up bet pays 35 to 1 in net profit, with the original $1 stake also returned.

Expected Value

E(X) = (1/37 × $35) + (36/37 × −$1)

= 35/37 − 36/37

E(X) = −$1/37 ≈ −$0.0270

House Edge

House edge = (1/37) × 100%

≈ 2.70%

The house advantage comes from the difference between the fair payout and the actual payout.

Fair Payout Comparison

The fair net profit on a $1 straight-up winning bet would be $36.

The actual net profit is $35.

That difference creates the house edge.

Comparing European and American Roulette

European roulette contains 37 pockets.

American roulette usually contains 38 pockets because it has both 0 and 00.

For a straight-up bet paying 35 to 1:

American Roulette Calculation

E(X) = (1/38 × $35) + (37/38 × −$1)

= −2/38

≈ −$0.0526

House edge:

≈ 5.26%

American roulette has a higher house advantage because of the additional pocket.

These figures assume standard straight-up bets and payouts.

Return to Player

Another way to describe gambling mathematics is return to player (RTP).

RTP represents the expected proportion of wagers returned to players as gross payouts over the long run.

RTP = Expected gross payout / Stake × 100%

For a game where all stakes are accounted for through payouts and operator profit:

RTP = 100% − House edge

Example

If a game has a house edge of 4%:

RTP = 100% − 4%

RTP = 96%

This means that, on average, the game returns $96 in gross payouts for every $100 wagered over a very large number of plays.

It does not mean a player who wagers $100 will necessarily receive $96.

Long-Term Expected Outcomes

Expected value becomes particularly important when a game is repeated many times.

Suppose a game has an expected loss of $0.50 per play.

If a player participates in n games, the expected total net profit is:

E(total profit) = n × E(profit per game)

This follows from the linearity of expectation and does not require independence, provided the expected value per play remains the same.

Example

Expected loss per game:

−$0.50

After 100 games:

E(total) = 100 × (−$0.50)

= −$50

After 1,000 games:

E(total) = 1,000 × (−$0.50)

= −$500

After 10,000 games:

E(total) = 10,000 × (−$0.50)

= −$5,000

The expected total loss grows with the amount of play.

The Law of Large Numbers

The Law of Large Numbers helps explain long-term behaviour in games of chance.

Under suitable conditions, the average result of many repeated independent games approaches the expected value.

For example, suppose a game has an expected loss of $0.20 per play.

After a small number of plays, the player's average result may be positive or negative.

After many plays, the average result per game tends toward −$0.20.

Important Distinction

The Law of Large Numbers does not guarantee that every player loses exactly the expected amount.

A player can experience a long winning streak.

However, repeated play does not change a negative expected value into a positive one.

Short-Term Luck Versus Long-Term Expectation

Consider a game in which a player:

• Wins $9 with probability 0.1.

• Loses $1 with probability 0.9.

Expected value:

E(X) = 0.1 × $9 + 0.9 × (−$1)

= $0.90 − $0.90

E(X) = $0

This is a fair game.

However, the results of ten plays may vary considerably.

One player might win three times and lose seven times:

Total profit = 3 × $9 − 7 × $1

= $20

Another player might lose all ten games:

Total profit = −$10

Both outcomes are possible even though the expected value is zero.

This demonstrates the difference between expected value and actual results.

Variance and Volatility in Games

Expected value describes the average outcome, but it does not describe how widely results can vary.

Variance and standard deviation measure this variability.

For a discrete random variable:

Var(X) = Σ[(x − μ)² × P(X = x)]

where:

μ = E(X)

Standard deviation:

σ = √Var(X)

Example: Two Fair Games

Game A:

• 50% chance of winning $1.

• 50% chance of losing $1.

Game B:

• 50% chance of winning $100.

• 50% chance of losing $100.

Both games have expected value zero.

However:

Game A has standard deviation $1.

Game B has standard deviation $100.

Game B has much greater variability and financial risk.

A fair game can still involve substantial losses.

Evaluating Gambling Strategies

Players sometimes use betting systems in an attempt to overcome the house advantage.

A common misconception is that changing the size or sequence of bets can turn a negative-expectation game into a profitable one.

However, if each wager has the same negative expected return per dollar, changing bet sizes does not remove the underlying disadvantage.

Example

Suppose a game has a house edge of 5%.

For a $10 wager:

Expected loss = $0.50

For a $100 wager:

Expected loss = $5

For a total of $1,000 wagered:

Expected loss = $50

Changing the distribution of wagers does not remove the expected loss associated with the total amount staked, assuming the same house edge applies.

The Martingale Strategy

The Martingale strategy is a betting system in which a player doubles the wager after each loss.

The idea is that a future win will recover earlier losses and produce a small profit.

Consider a simplified even-money game with a probability of winning slightly below 50%.

A player begins with a $1 bet.

After each loss, the next wager doubles.

The required wager increases rapidly.

Mathematical Problem

After k consecutive losses:

Next wager = 2ᵏ × Initial wager

The total amount lost is:

Total losses = (2ᵏ − 1) × Initial wager

After ten consecutive losses with a $1 initial wager:

Total losses = 2¹⁰ − 1

= $1,023

The next required wager would be:

$1,024

Why the Strategy Fails to Remove the House Edge

The strategy does not change the probability of winning an individual independent bet.

It also introduces substantial risks:

• Large wagers may be required.

• Players have limited funds.

• Games may have betting limits.

• Long losing sequences remain possible.

• A single losing sequence can erase many small gains.

The Martingale system changes the distribution of gains and losses but does not eliminate a negative expected value.

The Gambler's Fallacy

The gambler's fallacy is the mistaken belief that previous independent outcomes influence future probabilities.

Example: Coin Tosses

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

A player might believe that tails is now more likely.

However:

P(Tails on next toss) = 0.5

The probability remains unchanged.

Each toss is independent.

Roulette Example

Suppose a roulette wheel produces red several times in succession.

This does not make black more likely on the next spin, assuming independent spins and an unchanged wheel.

Past outcomes do not create a mathematical obligation for future outcomes to balance them.

Important Principle

Random processes do not compensate for previous outcomes simply because a particular result has occurred frequently or infrequently.

Hot Numbers and Cold Numbers

Some lottery players select numbers based on previous results.

A number that has appeared frequently may be called a hot number.

A number that has appeared infrequently may be called a cold number.

In a properly randomized lottery with independent draws:

• Previous appearances do not increase a number's probability.

• Previous absences do not increase a number's probability.

• Every valid combination has the same probability of being selected.

For example, the combination:

1, 2, 3, 4, 5, 6

has the same probability of being selected as any other specific six-number combination in a fair six-from-49 lottery.

Although one combination may look more unusual than another, their individual probabilities are equal.

Can Skill Change the Expected Value?

Not all gambling-related games are purely random.

Some games involve decisions that affect the probabilities or payouts.

For example:

• Certain card games involve strategic decisions.

• Some competitions involve both skill and chance.

• Promotional offers may change the effective cost of participation.

In these situations, mathematical analysis may need to account for the player's decisions and the rules of the game.

However, the existence of skill does not automatically imply a positive expected value.

The relevant probabilities, costs, and payouts must still be evaluated.

Practical Application: Prize Draws

Suppose a school organizes a fundraising prize draw.

There are 500 tickets.

Each ticket costs $4.

One ticket wins a prize worth $1,000.

Probability of Winning

P(win) = 1/500

Expected Gross Prize Value

E(gross prize) = 1/500 × $1,000

= $2

Expected Net Value

E(net) = $2 − $4

= −$2

The expected monetary loss per ticket is $2.

House Advantage

House edge = $2/$4 × 100%

= 50%

Interpretation

From a purely financial perspective, the ticket has a negative expected value.

However, participants may purchase tickets to support the school rather than to maximize financial return.

This illustrates how the purpose of an activity can influence how its value is interpreted.

Comparing Games Mathematically

Suppose three games each cost $5.

Game A

Expected gross payout:

1/2 × $8 = $4

Expected net profit:

$4 − $5 = −$1

House edge:

$1/$5 × 100% = 20%

Game B

Expected gross payout:

1/5 × $20 = $4

Expected net profit:

$4 − $5 = −$1

House edge:

20%

Game C

Expected gross payout:

1/10 × $40 = $4

Expected net profit:

$4 − $5 = −$1

House edge:

20%

Comparison

All three games have the same expected monetary loss.

However, they have different probabilities of winning and different prize sizes.

A player may experience very different short-term results depending on the game.

Expected value alone does not describe the complete risk.

Evaluating a Game of Chance

When analysing a game, follow a systematic process.

Identify the Outcomes

List the possible winning and losing outcomes.

Calculate the Probabilities

Determine the probability of each outcome.

Identify the Cost

Include the price of participation or amount wagered.

Determine Net Outcomes

Subtract the cost of playing from any gross payout.

Calculate Expected Value

Use:

E(X) = Σ[x × P(X = x)]

Determine Fairness

If E(X) = 0, the game is fair in expected monetary terms.

If E(X) is negative, the game favours the operator.

If E(X) is positive, the game favours the player under the model.

Calculate the House Edge

Use:

House edge = −E(X)/Stake × 100%

Evaluate Long-Term Outcomes

Consider how expected profit or loss changes with repeated play.

Consider Model Assumptions

Check whether outcomes are independent and whether probabilities and payouts have been represented correctly.

Common Mistakes

Confusing gross payout with net profit

The cost of playing must be included when calculating the player's expected profit.

Assuming a fair game guarantees breaking even

A fair game has zero expected net profit, but individual results can vary substantially.

Confusing odds with probability

Odds and probabilities describe related but different quantities.

Assuming a large jackpot makes a lottery favourable

A large prize must be evaluated relative to its probability and ticket cost.

Ignoring smaller prizes or shared jackpots

A complete lottery model must account for all relevant payouts and rules.

Assuming the house edge is the percentage lost on every bet

The house edge describes the expected long-run loss relative to wagers, not the outcome of every individual bet.

Believing a betting system changes independent probabilities

Changing the size of bets does not alter the underlying probability of winning.

Falling for the gambler's fallacy

Previous independent outcomes do not make the opposite outcome more likely.

Assuming past lottery numbers predict future numbers

Independent random draws do not depend on previous results.

Ignoring the possibility of severe losses

Games with similar expected values may have very different levels of risk.

Assuming the Law of Large Numbers guarantees short-term results

The law describes long-run convergence under appropriate conditions, not certainty over a small number of trials.

Key Terms

• Game of Chance: An activity in which random events influence the outcome.

• Lottery: A game involving randomly selected numbers or entries, usually with prizes.

• Probability: A measure of how likely an event is to occur.

• Sample Space: The set of all possible outcomes of a random experiment.

• Expected Value: The probability-weighted average of possible numerical outcomes.

• Expected Net Profit: The expected amount gained or lost after accounting for the cost of playing.

• Gross Payout: The total amount returned to a player when a prize is awarded.

• Net Profit: The gross payout minus the cost of participation.

• Fair Game: A game with an expected net profit of zero.

• Fair Payout: A payout that makes a game's expected net profit equal to zero.

• Fair Odds: Odds corresponding to a zero expected net profit.

• House Advantage: The operator's expected profit expressed as a percentage of the amount wagered.

• House Edge: Another term for house advantage.

• Return to Player (RTP): The expected percentage of wagers returned to players as gross payouts.

• Combination: A selection of objects in which order does not matter.

• Factorial: The product of all positive integers up to a specified positive integer.

• Variance: A measure of how widely outcomes are spread around their expected value.

• Standard Deviation: The square root of variance, measuring variability in the original units.

• Volatility: The degree to which results vary over time or between trials.

• Law of Large Numbers: A principle describing how averages from repeated trials approach expected values.

• Independent Events: Events where one outcome does not change the probability of another.

• Gambler's Fallacy: The mistaken belief that previous independent outcomes affect future probabilities.

• Martingale Strategy: A betting system that increases wagers after losses, commonly by doubling.

• Betting Limit: A restriction on the maximum or minimum amount that can be wagered.

• Long-Term Expectation: The average outcome predicted over many repetitions.

• Mathematical Advantage: A positive expected value for one participant relative to another.

• Risk: Exposure to uncertain outcomes, including possible losses.

• Probability Model: A mathematical representation of uncertain outcomes and their probabilities.

Key Takeaways

• Games of chance can be analysed using probability and expected value.

• Expected value describes the long-run average outcome of repeated play.

• Net profit must account for the cost of participating.

• A fair game has an expected net profit of zero.

• Fair payouts can be calculated using probabilities and participation costs.

• For a single-prize game, the fair gross payout is W = C/p.

• Lotteries often involve combinations because the order of selected numbers does not matter.

• Large jackpots do not necessarily make lottery tickets mathematically favourable.

• The house edge measures the operator's expected profit as a percentage of wagers.

• Return to player measures the expected percentage of wagers returned as gross payouts.

• European roulette has a standard straight-up house edge of approximately 2.70%.

• American roulette has a standard straight-up house edge of approximately 5.26%.

• Repeated play increases expected total losses in negative-expectation games.

• The Law of Large Numbers explains why long-run averages tend toward expected values.

• Short-term winning streaks can occur even in games with negative expected value.

• Variance and standard deviation help describe the financial risk of different games.

• Betting systems do not remove a negative house edge simply by changing wager sizes.

• The gambler's fallacy involves incorrectly assuming that independent events influence one another.

• Previous lottery outcomes do not predict future independent draws.

• Mathematical evaluation should consider probabilities, payouts, costs, assumptions, and long-term consequences.

Check Your Understanding

Recall and Understanding

1. Define a fair game and explain how expected value determines fairness.

2. Explain the difference between gross payout and net profit.

3. Define house advantage and describe how it is calculated.

4. Explain the Law of Large Numbers in the context of repeated gambling games.

5. Describe the gambler's fallacy and provide an example.

Application

6. A player pays $4 to roll a fair die. If a 6 appears, the player receives $18. Otherwise, the player receives nothing. Calculate the expected net profit.

7. A game costs $5 to play and has a 1/10 probability of winning. Calculate the fair gross payout.

8. A lottery ticket costs $3 and has a 1/1,000 probability of winning a single $2,000 prize. Calculate the expected net profit and house edge.

9. A roulette game has 37 equally likely pockets and pays 35 to 1 on a straight-up winning bet. Calculate the expected profit on a $10 wager.

10. A game has an expected loss of $0.75 per play. Calculate the expected total loss after 200 plays and 1,000 plays.

Analysis and Challenge

11. Two games have the same expected value but different prize sizes and winning probabilities. Explain why their levels of risk may differ.

12. A player uses a Martingale betting system, beginning with a $2 wager and doubling after every loss. Calculate the total amount lost after eight consecutive losses and the wager required for the ninth attempt.

13. A lottery requires selecting five numbers from 40. Calculate the probability of selecting the exact winning combination, assuming order does not matter.

14. A game costs $10. There is a 10% probability of winning $60, a 20% probability of winning $20, and a 70% probability of receiving nothing. Calculate the expected net profit and house edge.

15. Design a game of chance using dice, coins, or cards. Determine the possible outcomes and probabilities, establish a fair payout, introduce a house advantage of your choice, and explain the expected long-term outcome for players.

Final Summary

Games of chance provide practical applications of probability, expected value, combinations, and statistical reasoning.

Although individual results are unpredictable, the mathematical structure of a game allows us to calculate fair payouts, expected profits and losses, and house advantage.

A fair game has an expected net profit of zero, while a game with a negative expected value favours the operator over repeated play.

Lotteries and gambling games often offer large potential rewards, but these must be considered alongside their probabilities and the cost of participation.

The Law of Large Numbers helps explain why long-run average outcomes tend toward expected values, even though short-term results can vary considerably.

Understanding these principles also helps identify misconceptions such as the gambler's fallacy and the belief that betting systems can overcome a negative expected value.

The central mathematical lesson is that a game's attractiveness cannot be judged by its largest possible prize alone. Its probabilities, payouts, costs, and long-term expected outcomes must all be considered.