Probability Calculator
Calculate the probability of single events, union of events, intersection, conditional probability P(A|B), and complements.
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What Is a Probability Calculator?
A probability calculator computes the mathematical likelihood of an outcome occurring within a defined sample space. Ranging from 0 (impossible) to 1 (certainty), probability models everything from coin tosses and insurance risk underwriting to quantum mechanics and predictive artificial intelligence.
How to Use This Calculator
- Choose your calculation mode: Single Event, Union P(A ∪ B), Intersection P(A ∩ B), Conditional P(A|B), or Complement P(A').
- For single events, enter favorable outcomes and total sample space outcomes.
- For compound events, specify individual probabilities P(A) and P(B), and indicate whether events are mutually exclusive or independent.
- View the exact decimal probability, percentage, simplified fraction, and odds in favor and against.
Probability Rules & Axioms
P(A) = \frac{n(A)}{n(S)}, \quad P(A \cup B) = P(A) + P(B) - P(A \cap B), \quad P(A|B) = \frac{P(A \cap B)}{P(B)}Where n(A) is the count of favorable outcomes, n(S) is total sample space outcomes, P(A ∪ B) is the probability of either event occurring, and P(A|B) is the conditional probability of A given B has occurred.
Worked Example
Scenario: Calculate the probability of rolling a 2, 4, or 6 on a single fair 6-sided die.
Identify favorable outcomes: {2, 4, 6} → n(A) = 3.
Identify total possible outcomes: {1, 2, 3, 4, 5, 6} → n(S) = 6.
Apply the classical probability formula: P(A) = 3 / 6 = 0.50 (50.00%).
Determine odds: 1 to 1 (even odds).
Tips & Key Notes
- Probabilities can never be negative or exceed 1 (0% to 100%).
- Mutually exclusive events cannot occur simultaneously, meaning P(A ∩ B) = 0.
- If two events are statistically independent, P(A ∩ B) = P(A) × P(B) and P(A|B) = P(A).
- The complement rule states P(A') = 1 - P(A), often the fastest method to solve "at least one" probability questions.
Frequently Asked Questions
What is the difference between mutually exclusive and independent events?
Mutually exclusive events cannot happen at the same time (e.g., flipping heads and tails on a single coin flip). Independent events do not influence each other’s likelihood (e.g., flipping heads on coin #1 and rolling a 6 on die #2).
How do you convert probability into odds?
To convert probability P into odds in favor, divide P by (1 - P). For example, a 25% probability (0.25) gives odds in favor of 0.25 / 0.75 = 1 to 3 (1:3).
What does conditional probability P(A|B) mean?
Conditional probability measures the probability of event A occurring given that event B is already known to have occurred. It shrinks the effective sample space to outcome B.
What is the law of total probability?
The law of total probability states that if events B1, B2, ..., Bn partition a sample space, the total probability of an event A is the sum of P(A|Bi) × P(Bi) across all partitions.
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