What is the Gambler's Fallacy?

The Gambler's Fallacy is the incorrect belief that if an independent event happens frequently in the past, it is less likely to happen in the future. For example, if a roulette wheel lands on black five times in a row, the probability of landing on black the sixth time is still exactly 47.4%. The coin has no memory.

What is the difference between Independent and Dependent events?

An independent event is unaffected by previous trials (e.g., flipping a coin). A dependent event changes based on what just happened. If you draw an Ace from a deck of 52 cards and do not put it back, the probability of drawing another Ace drops because there are now only 3 Aces left in a 51-card deck.

How do I calculate the probability of multiple independent events happening?

You multiply their individual probabilities. To flip 'Heads' (50% or 0.5) three times in a row, the math is 0.5 * 0.5 * 0.5 = 0.125. There is a 12.5% chance of three consecutive Heads.

Does Kodivio track my inputted data?

No. The Kodivio Probability engine operates strictly via client-side Javascript. Mathematical logic is executed securely within your local browser, meaning zero data is sent to external servers.

Probability Calculator

Quantify Statistical Uncertainty, Combinatorics, and Independent Event Odds.

Combinatorial Logic

Zero Latency Execution

100% Local Math Engine

Two Event Probability

Probability of Event A (0 to 1)

Probability of Event B (0 to 1)

A AND B Occur0.2500

A OR B Occur0.7500

Binomial Probability (Successes in Trials)

Number of Trials (n)

Exact Successes (k)

Likelihood of exactly 5 successes0.2461

Mastering Statistical Uncertainty

The universe is governed by mathematical uncertainty. From actuarial risk assessment in the insurance industry to predictive modeling in quantum physics, the ability to accurately calculate the likelihood of an event is one of the most powerful tools in modern science.

The Kodivio Exact Probability Calculator is engineered to eliminate human cognitive bias. Utilizing a local, high-precision Javascript floating-point engine, our tool allows students, researchers, and data analysts to instantly compute complex fractional odds, independent event sequencing, and mutually exclusive outcomes without relying on manual long-form division.

The Rules of Independent vs. Dependent Variables

In combinatorics, the most critical question to ask before running a calculation is, "Does the first event affect the environment of the second event?"

Rule of Independent Events

Also known as Events with Replacement. If you pull a marble from a bag of 10, look at its color, and put it back before pulling again, the denominator (10) never changes. A coin flip is the ultimate independent event.

P(A and B) = P(A) × P(B)

Rule of Dependent Events

Also known as Events without Replacement. If you pull a marble from a bag of 10 and keep it, the next time you reach in, the denominator has shifted to 9. The statistical environment has permanently changed.

P(A and B) = P(A) × P(B | A)

Overcoming Human Bias: The Gambler's Fallacy

The human brain is evolutionarily hardwired to find "patterns" in random data, which leads to a severe cognitive failure known as The Gambler's Fallacy (or the Monte Carlo Fallacy).

In 1913, at the Monte Carlo Casino, the roulette wheel landed on black 26 consecutive times. Because patrons assumed that a "red" outcome was statically "due" to balance out the universe, millions of francs were lost betting against the streak. The mathematical reality is that wheel has no memory. On spin 27, the probability of hitting black was still exactly 47.4% (on a European wheel with one green zero), entirely unaffected by the previous 26 spins. Our calculator prevents this emotional bias by forcing you to look at raw, isolated integers.

Frequently Asked Questions

Why does probability always formulate between 0 and 1?

In mathematics, a probability of `0` means biological or physical impossibility, while a probability of `1.0` is an absolute certainty. A 50% chance is written algebraically as 0.50. You cannot have a 110% chance of an event physically occurring.

What does 'Mutually Exclusive' mean?

Two events are mutually exclusive if they cannot happen at the same time. You cannot roll a standard 6-sided die and have it land on '2' and '5' simultaneously. Therefore, the probability of rolling a 2 OR a 5 is calculated using the Addition Rule: 1/6 + 1/6 = 33.3%.