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Probability Calculator

Probability measures the likelihood of an event on a scale from 0 (impossible) to 1 (certain). For a single event, P(E) = favorable outcomes / total outcomes. For two independent events, P(A and B) = P(A) * P(B), and P(A or B) = 1 - (1 - P(A))(1 - P(B)). Conditional probability is P(A|B) = P(A and B) / P(B). This calculator covers all three scenarios: single event probability with odds, combined independent events using AND/OR logic, and conditional probability using Bayes' definition.

Quick Answer

Drawing one ace from a standard 52-card deck has probability 4/52 = 0.0769, or about 7.69%, with odds of 4:48 (simplified 1:12).

Single Event Probability

Combined Independent Events

Conditional Probability P(A|B)

Common Examples

Input Result
4 favorable out of 52 total P = 0.0769 (7.69%), odds 1:12
P(A) = 0.5, P(B) = 0.5 (AND) P(A and B) = 0.25 (25%)
P(A) = 0.5, P(B) = 0.5 (OR) P(A or B) = 0.75 (75%)
P(A and B) = 0.12, P(B) = 0.4 P(A|B) = 0.3 (30%)
1 favorable out of 6 total P = 0.1667 (16.67%), odds 1:5

How It Works

Single Event Probability

The probability of a single event is the ratio of favorable outcomes to total possible outcomes:

P(E) = favorable outcomes / total outcomes

This value always falls between 0 and 1. A probability of 0 means the event is impossible; a probability of 1 means it is certain.

The complement of an event is the probability it does NOT occur: P(not E) = 1 - P(E).

Odds express probability as a ratio of favorable to unfavorable outcomes. If P(E) = 1/6, the odds in favor are 1:5 (one favorable outcome for every five unfavorable).

Combined Independent Events

Two events are independent if the occurrence of one does not affect the probability of the other (such as two separate coin flips).

AND (intersection): P(A and B) = P(A) * P(B). The probability that all events occur is the product of their individual probabilities.

OR (union): P(A or B) = 1 - (1 - P(A)) * (1 - P(B)). For independent events, the probability that at least one occurs equals one minus the probability that none occur.

Conditional Probability

**P(A B)** is the probability of event A given that event B has already occurred:
**P(A B) = P(A and B) / P(B)**

This formula requires P(B) > 0. Conditional probability is the foundation of Bayes’ theorem and is central to statistics, machine learning, and decision-making.

Worked Example

A standard deck has 52 cards. The probability of drawing an ace is P = 4/52 = 1/13 = 0.0769 (about 7.69%). The complement is 1 - 0.0769 = 0.9231. The odds of drawing an ace are 4:48, which simplifies to 1:12.

For two independent coin flips (each P = 0.5): P(both heads) = 0.5 * 0.5 = 0.25. P(at least one head) = 1 - (0.5)(0.5) = 0.75.

For conditional probability: if 12% of customers buy both product A and B, and 40% buy product B, then P(A B) = 0.12 / 0.40 = 0.30, meaning 30% of product B buyers also buy product A.

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Frequently Asked Questions

What is the difference between probability and odds?
Probability is the ratio of favorable outcomes to total outcomes (e.g., 1/6 for rolling a specific number on a die). Odds compare favorable to unfavorable outcomes (e.g., 1:5 for the same event). Probability ranges from 0 to 1, while odds range from 0:n to n:0.
What does 'independent' mean in probability?
Two events are independent if the occurrence of one has no effect on the probability of the other. Flipping a coin twice produces independent events, because the first flip does not change the probability of the second. Drawing cards without replacement produces dependent events, because each draw changes the deck composition.
How is conditional probability different from regular probability?
Regular probability P(A) considers all possible outcomes. Conditional probability P(A|B) restricts the sample space to only those outcomes where B has occurred, then asks how likely A is within that restricted space. It answers the question: given that B happened, what is the probability of A?
Can probability be greater than 1?
No. Probability always falls between 0 (impossible) and 1 (certain), inclusive. If a calculation produces a value outside this range, the inputs are invalid. Percentages can reach 100% but represent the same 0-to-1 scale.
What is the complement of a probability?
The complement of P(E) is P(not E) = 1 - P(E). It represents the probability that the event does NOT occur. Complements are useful for calculating 'at least one' scenarios: P(at least one) = 1 - P(none).