A "Fair" Gamble

In exploring the probabilities behind classic casino card games like Blackjack, Craps, and Poker, we begin by understanding the foundational concept of a "fair game" through simple card-drawing scenarios. These examples highlight the core principles that determine fairness in games involving chance.

A fair game is one where, over time, a player neither gains nor loses money on average. This means the expected value—the average amount a player can expect to win or lose per play—is zero. Achieving this balance requires two key conditions:

1. Equal Likelihood of Outcomes: Each possible outcome (e.g., drawing a card of any suit) must have an equal probability. For example, in a standard 52-card deck, each suit (hearts, clubs, diamonds, spades) has exactly 13 cards, so the chance of drawing any particular suit is equal (1/4).
2. Balanced Payoffs: The rewards and losses associated with each outcome must balance out so that the total expected gain equals the total expected loss. For instance, if you earn $2 for drawing hearts and $4 for diamonds, but lose $1 for clubs and $5 for spades, the total gains ($2 + $4 = $6) equal the total losses ($1 + $5 = $6). Because the probabilities are equal, this ensures the game is fair.

Let's illustrate with an few examples:

• Drawing a Jack or a Diamond: When calculating the probability of drawing a Jack or a Diamond, it’s important to avoid double-counting cards that satisfy both conditions (like the Jack of Diamonds). This careful counting ensures accurate probabilities, a prerequisite for fairness.
• Drawing Multiple Cards Without Replacement: The probability of drawing four aces from five cards depends on which card is left out, showing how conditional probabilities affect outcomes and fairness.
• Matching Suits in Two Draws: The probability of drawing two cards of the same suit depends on the composition of the deck after the first draw, illustrating how sequential events influence fairness calculations.
• Comparing Scenarios for Pair Probability: When two players are dealt cards, analyzing how many pairs remain in the deck helps determine which scenario is more likely to yield a pair, emphasizing the role of combinatorics in assessing fairness.

A fair card game balances the mathematics of probability with the economics of payout. By ensuring equal chances for all outcomes and matching rewards to these probabilities, the game neither favors the player nor the house in the long run. This foundational understanding is critical before diving deeper into the complexities of casino games like Blackjack, Craps, and Poker.

Dice games, like card games, rely on chance and probability. Understanding what makes a dice game fair involves analyzing the likelihood of outcomes and ensuring the rewards balance the risks. Here, we explore these concepts through classic dice scenarios and generalize the principles behind fairness. A standard six-sided die (d6) has six equally likely outcomes, each with a probability of $$\frac{1}{6}$$. When rolling two dice, there are $$6 \times 6 = 36$$ equally likely outcomes because each die is independent. For example, the probability that both dice show the same number (doubles) is:

$$\frac{6}{36} = \frac{1}{6}$$

since there are 6 doubles (1-1, 2-2, …, 6-6) out of 36 total outcomes.

A game is fair if the expected value (average payoff) for the player equals the cost of playing. Consider a game where:

• You pay a fixed amount to roll a die.
• You receive a payout depending on the outcome.

For instance, if you pay $3 to roll a die and receive a payout equal to the number rolled (1 through 6), the expected value per roll is:

$$\frac{1}{6} \times (1 + 2 + 3 + 4 + 5 + 6) = \frac{21}{6} = 3.5$$

Since you pay $3 but expect to receive $3.50 on average, this game favors the player and is thus rationally, making bets on this dice game would be justifiable risk-seeking.

Fairness also depends on the structure of bets and payouts. For example:

• Bet A: Earn $4 if the die shows 1, 2, or 3; otherwise, earn nothing. Expected value: $$\frac{3}{6} \times 4 = 2$$
• Bet B: Earn $10 if the die shows 6; otherwise, nothing. Expected value: $$\frac{1}{6} \times 10 \approx 1.66$$

Bet A has a higher expected value and is better for the player in the long run.

Similarly, we can go through some examples with two dice:

• Bet A: Earn $1 if the sum is 6 or 8 (10 ways out of 36). Expected value: $$\frac{10}{36} \times 1 \approx 0.28$$
• Bet B: Earn $10 if the sum is 12 (1 way out of 36). Expected value: $$\frac{1}{36} \times 10 \approx 0.28$$

Both bets have the same expected value, but Bet A offers more frequent smaller wins, while Bet B offers a rare big win. In reality, no dealer or casino would facilitate games like the examples above.

A fair dice game balances the mathematics of probability with the economics of payouts. By ensuring equal likelihood of outcomes and matching expected winnings to the cost of play, the game offers neither an advantage nor disadvantage to the player over time. Understanding these principles helps players and designers create and evaluate dice games that are equitable and engaging.

From these examples, we derive general principles that make a game fair:

• Accurate Probability Accounting: Correctly identifying and calculating probabilities without overlap or omission is essential.
• Equal Probability Distribution: All possible outcomes should have equal or well-understood probabilities to avoid bias.
• Balanced Reward Structure: The payout scheme must be designed so that expected winnings equal expected losses over time.
• Transparency and Consistency: The rules and payouts should be clear and consistently applied to maintain fairness.

Winning or losing streaks are common phenomena where players experiences several consecutive wins or losses. Importantly, anyone can get lucky or unlucky a few times in a row purely by chance, without any underlying change in skill or conditions. This means that short streaks, even of several games, can occur naturally in random sequences of outcomes. With that said, Streak Probability can be estimated using Feller’s Approximation.

To estimate the probability of observing at least one streak of length $$n$$ in $$N$$ trials, where each trial has a probability $$p$$ of success (or failure, depending on context), Feller’s approximation is used:

$$P \approx 1 - (1 - p^n)^{N - n + 1}$$

$$P$$ is the probability of streak of success or failure $$n$$ consecutive times, happening at least once in $$N$$ trials.

Mathematical Proof

• Define the event: We want the probability of at least one streak of length $$n$$ occurring in $$N$$ independent trials.
• Probability of a streak starting at a specific position: The probability that a streak of length $$n$$ starts at trial $$i$$ is $$p^n$$ (assuming independent trials and constant probability $$p$$).
• Number of possible starting points: There are $$N - n + 1$$ possible starting positions for such a streak within $$N$$ trials.
• Probability of no streak at a given position: The probability that a streak does not start at position $$i$$ is $$1 - p^n$$.

• Assuming independence of streak starts: The probability that no streak of length $$n$$ starts anywhere in the $$N$$ trials is approximately:

  $$(1 - p^n)^{N - n + 1}$$

• Therefore, the probability of at least one streak of length $$n$$ is:

  $$P = 1 - (1 - p^n)^{N - n + 1}$$

  This formula provides an approximation because it assumes independence between streak-start events, which is not strictly true, but it is a useful and widely accepted estimate.

Statistical Considerations and Limitations

• Approximation Nature: Feller’s formula is an approximation. In practice, the assumption of independence between streak-start events is violated because streaks can overlap and influence each other.
• Margin of Error: Due to this, there is a margin of error in the estimated probability. The actual probability might differ, especially for small $$N$$ or large $$n$$.
• Hypothesis Testing: To determine if observed streaks are statistically significant (i.e., unlikely to be due to chance), proper hypothesis testing should be performed. This involves comparing observed streak frequencies with expected frequencies under the null hypothesis of randomness.
• Practical Implications: Without such testing, concluding that a streak indicates a change in skill, strategy, or external factors is premature. Many apparent streaks can be explained by random chance alone.

While streaks may feel meaningful, their occurrence is often consistent with probability theory and chance, and Feller’s approximation offers a practical way to estimate their likelihood, albeit with cautions.

The Kelly Criterion is a mathematical formula developed in 1956 by John L. Kelly Jr., a scientist at Bell Labs. Initially it was designed to optimize long-distance telephone signal noise, though it gained prominence in gambling and investing for determining the optimal bet size to maximize long-term wealth growth.

The Kelly formula calculates the fraction of capital to allocate to a bet or investment:

$$f^* = \frac{bp - q}{b}$$

Where:

• $$f^*$$ = Optimal fraction of capital to bet
• $$b$$ = Net odds received (e.g., $$b = 2$$ for 2:1 odds)
• $$p$$ = Probability of winning
• $$q = 1 - p$$ = Probability of losing

It provides practical applications in investing, gambling and risk management:

• Investing: Stock selection, portfolio balancing, and asset allocation (e.g., stocks, bonds, real estate).
• Gambling: Horse racing, sports betting, and casino games.
• Risk Management: Prevents overexposure by capping bet sizes based on edge and odds.

Let's go through a Step-by-Step Example using Stock Investment. Suppose you analyze a tech stock with:

• 60% chance of a 20% return ($$p = 0.6$$, $$b = 0.2$$)
• 40% chance of a 10% loss ($$q = 0.4$$, $$a = 0.1$$)
• Note that sum of $$p$$ and $$q$$ is always 1 or 100%

Kelly Calculation: $$f^* = \frac{(0.2 \times 0.6) - 0.4}{0.2} = \frac{0.12 - 0.4}{0.2} = -1.4$$

A negative result means no bet. Adjust assumptions:

• If the loss is 5% instead of 10% ($$b = 0.2 / 0.05 = 4$$):

$$f^* = \frac{(4 \times 0.6) - 0.4}{4} = \frac{2.4 - 0.4}{4} = 0.5 \quad (50\% \text{ of capital})$$

Mathematical Proof

The objective is to maximize the geometric growth rate of wealth.

Expected Logarithmic Wealth

After $$n$$ bets, wealth $$G_n$$ grows as:

$$G_n = (1 + bf)^W \times (1 - f)^L$$

where $$W$$ = wins, $$L$$ = losses.

Maximize Growth Rate

Take the logarithm and expected value:

$$\mathbb{E}[\log G] = p \log(1 + bf) + q \log(1 - f)$$

Derivative and Optimization

Differentiate with respect to $$f$$:

$$\frac{d}{df} \mathbb{E}[\log G] = \frac{pb}{1 + bf} - \frac{q}{1 - f} = 0$$

Solve for $$f^*$$:

$$\frac{pb}{1 + bf^*} = \frac{q}{1 - f^*}$$

Cross-multiplied:

$$pb(1 - f^*) = q(1 + bf^*)$$

Simplify:

$$pb - pbf^* = q + qbf^*$$

$$pb - q = f^*(pb + qb)$$

$$f^* = \frac{pb - q}{b(p + q)} = \frac{pb - (1 - p)}{b} = \frac{p(b + 1) - 1}{b}$$

For binary outcomes ($$p + q = 1$$):

$$f^* = \frac{bp - q}{b}$$

The Kelly Criterion balances growth and risk by mathematically deriving the optimal bet size. While powerful, it assumes precise probability and odds estimates, making conservative "fractional Kelly" strategies (e.g., betting half of $$f^*$$) common in practice.