Probability Calculator
Calculate basic probability, compound probability (AND/OR), and conditional probability with Bayes' theorem. Get instant results with odds, percentages, and step-by-step probability breakdowns.
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Probability Formulas
What is Probability?
Probability is a branch of mathematics that measures the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain). The basic probability formula is P(A) = favorable outcomes / total possible outcomes. For example, the probability of rolling a 3 on a fair die is 1/6, because there is 1 favorable outcome out of 6 total possible outcomes.
Probability theory forms the foundation of statistics, data science, machine learning, risk assessment, and decision-making across virtually every field. Understanding probability helps you evaluate uncertainty, make informed predictions, and assess the likelihood of outcomes in games, experiments, financial models, medical diagnostics, and everyday decisions.
There are three main types of probability calculations: basic probability (single event), compound probability (combining two or more events using AND/OR operations), and conditional probability (the probability of an event given that another event has occurred, using Bayes' theorem).
Common Probability Examples
Understanding probability through familiar real-world examples makes abstract concepts concrete and intuitive.
Coin Toss
A fair coin has two equally likely outcomes: heads or tails. P(Heads) = 1/2 = 0.5 = 50%.
Two heads in a row: P = (1/2) × (1/2) = 1/4 = 25% (independent events, multiply)
At least one head in two flips: P = 1 - P(no heads) = 1 - (1/2)² = 3/4 = 75%
Three heads in a row: P = (1/2)³ = 1/8 = 12.5%
Dice Roll
A standard six-sided die has 6 equally likely outcomes (1 through 6).
Rolling a 6: P = 1/6 ≈ 16.67%
Rolling even (2, 4, or 6): P = 3/6 = 1/2 = 50%
Rolling > 4 (5 or 6): P = 2/6 = 1/3 ≈ 33.33%
Two dice summing to 7: P = 6/36 = 1/6 ≈ 16.67% (6 favorable combinations)
Card Drawing
A standard deck has 52 cards: 4 suits × 13 ranks. Card probabilities change when drawing without replacement (dependent events).
Drawing an Ace: P = 4/52 = 1/13 ≈ 7.69%
Drawing a Heart: P = 13/52 = 1/4 = 25%
Drawing Ace of Spades: P = 1/52 ≈ 1.92%
Two Aces in a row (no replacement): P = (4/52) × (3/51) ≈ 0.45%
Real-World Probability
Birthday Problem: In a group of 23 people, there's a ~50% chance two share a birthday
Lottery (6 from 49): P ≈ 1 in 13,983,816 (0.0000072%)
Medical test accuracy: A 99% accurate test can still give surprising results due to base rate fallacy—use Bayes' theorem to evaluate
Understanding Probability Types
1. Basic (Classical) Probability
Used when all outcomes are equally likely. P(A) = number of favorable outcomes / total number of outcomes. This applies to fair coins, dice, cards, and random selections. The probability always falls between 0 (impossible) and 1 (certain). The complement rule states P(NOT A) = 1 - P(A), which is often the easiest way to solve "at least one" problems.
2. Compound Probability
Combines two or more events. The multiplication rule (AND): P(A ∩ B) = P(A) × P(B) for independent events. The addition rule (OR): P(A ∪ B) = P(A) + P(B) - P(A ∩ B). For mutually exclusive events (events that cannot both occur), P(A ∩ B) = 0, so P(A OR B) simplifies to P(A) + P(B). Key concept: determine whether events are independent or dependent before calculating.
3. Conditional Probability (Bayes' Theorem)
The probability of event A occurring given that event B has already occurred: P(A|B) = P(A ∩ B) / P(B). Bayes' Theorem reverses this relationship: P(A|B) = P(B|A) × P(A) / P(B). This is crucial in medical testing (false positive rates), machine learning (Naive Bayes classifier), spam detection, and updating beliefs with new evidence.
Frequently Asked Questions
What does P(A AND B) vs P(A OR B) mean?
P(A AND B) is the probability that both events happen simultaneously. P(A OR B) is the probability that at least one of the events happens. AND is more restrictive (resulting in smaller probability), while OR is less restrictive (resulting in larger probability). For independent events, P(A AND B) = P(A) × P(B), while P(A OR B) = P(A) + P(B) - P(A AND B).
When are events independent?
Events are independent when the occurrence of one doesn't affect the probability of the other. For example, coin flips are independent—getting heads on the first flip doesn't change the probability of heads on the second. Drawing cards without replacement are dependent events because removing a card changes the composition of the remaining deck.
What is Bayes' Theorem and when do I use it?
Bayes' Theorem calculates the probability of an event based on prior knowledge of conditions related to the event. The formula is P(A|B) = P(B|A) × P(A) / P(B). Use it when you know the probability of B given A, but need to find the probability of A given B—common in medical testing, spam filtering, and data science.
How do I calculate the probability of rolling a specific number on a die?
For a fair six-sided die, each face has an equal probability of 1/6 (approximately 0.1667 or 16.67%). To find the probability of rolling a specific number, use P = favorable outcomes / total outcomes = 1/6. For rolling any one of multiple numbers (e.g., 1 or 2), add the individual probabilities: 2/6 = 1/3.
What is the probability of getting heads on a coin toss?
For a fair coin, the probability of heads is 1/2 = 0.5 = 50%. For multiple independent coin tosses, the probability of getting heads on all tosses is (1/2)^n, where n is the number of tosses. For example, getting heads twice in a row: (1/2)² = 1/4 = 25%.
How do I convert between probability, percentage, and odds?
Probability is expressed as a number between 0 and 1. Percentage = Probability × 100. Odds in favor = favorable : unfavorable (e.g., probability 0.25 = 25% = 1:3 odds). To convert odds (a:b) back to probability: P = a / (a + b). For example, 3:2 odds = 3/(3+2) = 0.6 = 60%.