Probability Calculator
Free probability calculator. Calculate the probability of single events, multiple independent events, and complement events with clear examples.
Probability Calculator
P(Event) = Number of favorable outcomes ÷ Total possible outcomes
Comprehensive Guide: Theoretical, Experimental & Conditional Probability
Master probability axioms, independent vs. dependent events, conditional probability, Bayes theorem, and the Gambler’s Fallacy.
1. Mathematical Foundations & Kolmogorov Axioms
In probability theory, the likelihood of an event E occurring within a sample space S is quantified on a continuous closed scale from 0 (impossible) to 1 (certain):
0 ≤ P(E) ≤ 1 | P(E') = 1 − P(E) (Complementary Probability)
2. Compound Probability Rules: Addition & Multiplication Rules
When analyzing multiple intersecting or sequential events:
- Addition Rule (Union 'OR'):
P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If mutually exclusive (cannot happen together),P(A ∩ B) = 0. - Multiplication Rule (Intersection 'AND'): For independent events,
P(A ∩ B) = P(A) × P(B). For dependent events,P(A ∩ B) = P(A) × P(B|A).
3. Graduated Step-by-Step Worked Problems
Problem 1: Drawing Cards with Non-Mutually Exclusive Events
What is the probability of drawing either a Heart OR a King from a standard shuffled 52-card deck?
Step 1: Total sample space: n(S) = 52.
Step 2: Probability of Heart: P(Heart) = 13 / 52.
Step 3: Probability of King: P(King) = 4 / 52.
Step 4: Probability of King of Hearts (overlap): P(Heart ∩ King) = 1 / 52.
Step 5: Apply Addition Rule: P = 13/52 + 4/52 − 1/52 = 16/52 = 4/13 ≈ 0.3077 (30.77%).
Result: Probability is 4/13 (30.8%).
Problem 2: Independent Sequential Coin Tosses
What is the probability of flipping a fair coin and getting 'Heads' 5 consecutive times?
• Each flip is independent: P(H) = 0.5.
• P(5 Heads) = (1/2)⁵ = 1/32 = 0.03125 (3.125%).
• Odd against: 31 to 1.
Problem 3: Sampling Without Replacement (Conditional / Dependent Probability)
An urn contains 5 red marbles and 7 blue marbles (12 total). Two marbles are drawn sequentially without replacement. What is the probability that both are red?
• First draw: P(R₁) = 5 / 12.
• Second draw: 4 red marbles remain out of 11 total: P(R₂|R₁) = 4 / 11.
• Joint probability: P(R₁ ∩ R₂) = (5/12) × (4/11) = 20 / 132 = 5 / 33 ≈ 0.1515 (15.15%).
Result: Probability is exactly 5/33 (15.2%).
4. The Gambler’s Fallacy: The Illusion of Due Outcomes
A widespread psychological trap in gaming is the Gambler’s Fallacy—believing that past independent events influence future probabilities (e.g. "Red has hit 6 times in a row on roulette, so Black is overdue!").
Each spin of a balanced roulette wheel or toss of a fair coin has zero memory of prior outcomes. The probability of Black remains strictly 18/38 ≈ 47.37% on an American wheel, regardless of previous streaks.
5. Frequently Asked Questions (FAQ)
What is the difference between odds and probability?
Probability is the ratio of favorable outcomes to total outcomes (Favorable / Total). Odds is the ratio of favorable outcomes to unfavorable outcomes (Favorable : Unfavorable). A 20% probability (1 in 5 chance) equals odds of 1 to 4.
What is Bayes' Theorem?
Bayes' Theorem computes conditional probability: P(A|B) = [P(B|A) · P(A)] / P(B). It updates the probability of a hypothesis (A) given new evidence (B), central to medical diagnostics and machine learning spam filters.
Can an event with probability 0 still happen?
In continuous probability distributions, the probability of selecting any single exact real number (such as an exact height of 175.000000... cm) is mathematically zero (P = 0), yet every real outcome observed was one of those zero-probability points.