Casino Korea

Probability Calculator

Understanding probability is fundamental to comprehending how casino games work and why the house always maintains a mathematical advantage. This calculator helps you explore true odds, expected value, and multi-trial probabilities in gambling scenarios.

Probability in gambling determines the likelihood of specific outcomes occurring. Casinos exploit the gap between true odds (actual probability) and payout odds (what they pay winners) to ensure long-term profitability. This mathematical foundation, as explained by the Wolfram MathWorld probability reference, is what makes gambling fundamentally unfavorable for players over time.

Calculate Probability and True Odds

Enter the number of winning outcomes and total possible outcomes to calculate probability and true odds.

Quick Presets:

Number of outcomes that result in a win
Total number of possible outcomes

Results

Probability:
Decimal Probability:
Probability expressed as a decimal (0 to 1)
True Odds (Against):
For every 1 win, expect this many losses on average
Implied Fair Payout:
What the payout should be if there were no house edge

Calculate Expected Value

Determine the mathematical expectation for a bet based on probability and payout.

Probability of winning the bet
1 = even money, 35 = 35 to 1

Expected Value Results

Expected Value (per bet):
House Edge:
The casino's mathematical advantage on this bet
Expected Return:
For every $100 wagered, expect to receive this amount back
Expected After 100 Bets:

Multi-Trial Probability

Calculate the probability of winning at least once (or a specific number of times) over multiple attempts.

Probability of winning one trial
How many times you'll attempt

Multi-Trial Results

Probability:
Odds:
Expected Wins:
Average number of wins over this many trials

Understanding Probability in Casino Games

Probability forms the mathematical foundation of all casino games. It expresses the likelihood of an outcome occurring as a number between 0 (impossible) and 1 (certain), or equivalently as a percentage between 0% and 100%. According to the Stanford Encyclopedia of Philosophy's overview of probability theory, these principles have been studied mathematically since the 17th century when Blaise Pascal and Pierre de Fermat analyzed games of chance.

The Relationship Between Probability and Odds

Probability and odds express the same concept differently. If an event has a probability of 25% (0.25), the true odds against it are 3 to 1—meaning you expect 3 losses for every 1 win. Understanding this distinction is crucial because casinos pay out at lower odds than the true odds warrant, creating their mathematical advantage.

Probability = Winning Outcomes ÷ Total Outcomes
True Odds (Against) = (Total - Winning) : Winning
Fair Payout = (1 ÷ Probability) - 1

Common Casino Game Probabilities

Game / Bet Probability True Odds Casino Pays House Edge
Roulette - Single Number (American) 2.63% 37 to 1 35 to 1 5.26%
Roulette - Red/Black (American) 47.37% 1.11 to 1 1 to 1 5.26%
Craps - Pass Line 49.29% 1.03 to 1 1 to 1 1.41%
Baccarat - Banker 45.86% 1.18 to 1 0.95 to 1 1.06%
Blackjack - Natural 21 4.83% 19.7 to 1 1.5 to 1 Varies
Keno - Match 1 of 1 25.00% 3 to 1 3 to 1 25%+

Expected Value: The Mathematical Reality

Expected value (EV) is the most important concept in gambling mathematics. It represents the average outcome of a bet if it were repeated infinitely many times. A negative expected value means the player loses money over time; a positive expected value means the player profits. In casino games, the expected value is almost always negative for the player.

The Khan Academy's explanation of expected value provides an excellent foundation for understanding this concept, which applies not only to gambling but to decision-making under uncertainty in general.

Expected Value = (Win Probability × Win Amount) - (Loss Probability × Loss Amount)

Example: American Roulette Even Money Bet

Betting $100 on red in American roulette:

  • Win probability: 18/38 = 47.37%
  • Loss probability: 20/38 = 52.63%
  • Win amount: $100
  • Loss amount: $100

EV = (0.4737 × $100) - (0.5263 × $100) = $47.37 - $52.63 = -$5.26

This means for every $100 bet on red, you expect to lose $5.26 on average—the 5.26% house edge.

The Gambler's Fallacy

One of the most common misconceptions about probability is the gambler's fallacy—the belief that past outcomes influence future independent events. If a roulette wheel lands on red 10 times in a row, the probability of the next spin being black is still exactly 47.37% on an American wheel. Each spin is independent.

Research published by the Journal of Behavioral Addictions (NIH/PubMed) demonstrates how the gambler's fallacy and related cognitive distortions contribute to problem gambling behaviors. Understanding that probability doesn't have "memory" is essential for realistic expectations about gambling outcomes.

Multi-Trial Probabilities

While a single bet on a roulette number has only a 2.63% chance of winning, the probability of winning at least once increases with more attempts. However, this doesn't mean you'll come out ahead—each losing bet costs money, and the house edge compounds over time.

The probability of at least one win in n independent trials is:

P(at least 1 win) = 1 - (1 - p)^n

Where p = probability of winning a single trial
And n = number of trials

For example, betting on a single number 38 times (one complete cycle of American roulette outcomes) gives a 63.4% probability of winning at least once. But this doesn't guarantee profit—you'll wager $38 to potentially win $35 plus your $1 bet back, while losing on 37 other bets.

Connection to South Korean Gambling Law

South Korea's restrictive approach to gambling, detailed in our legal framework section, stems partly from understanding these mathematical realities. The government recognizes that negative expected value in casino games ensures systematic wealth transfer from players to operators, contributing to gambling addiction and financial hardship.

At Kangwon Land, South Korea's only legal casino for Korean citizens, all games operate with standard probability structures that comply with international gaming regulations. These probabilities are regularly audited to ensure fairness within the established mathematical parameters—meaning the games are "fair" in that they operate as designed, but the design inherently favors the house.

For those affected by gambling problems, our responsible gambling resources page provides information about support organizations and treatment options.

Educational Purpose

This calculator is provided solely for educational purposes to help individuals understand the mathematical principles underlying casino games. It demonstrates why gambling should be viewed as paid entertainment with a known cost (the house edge) rather than a potential income source.

Understanding probability helps explain why casinos are profitable businesses. The mathematics are transparent and well-established—no secret systems or strategies can overcome the fundamental reality that casino games have negative expected value for players.

Important Reminder

This calculator provides educational information about gambling mathematics. It cannot and should not be used to develop betting strategies or systems, as no strategy can overcome the mathematical house edge built into casino games. All forms of gambling carry risk of financial loss. If you or someone you know has a gambling problem, seek professional help immediately.

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