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Probability Calculator

Calculate probabilities using basic rules and common distributions (binomial, normal, Poisson) with interactive visualizations and detailed explanations.

Probability Calculator

Probability of event A

Probability of event B

Probability of A and B

Probability Results

P(A) = 0.3, P(B) = 0.4, P(A∩B) = 0.1

Union: P(A ∪ B)
0.6000
Probability of A or B
Intersection: P(A ∩ B)
0.1000
Probability of A and B
P(A|B)
0.2500
A given B
P(B|A)
0.3333
B given A
Independence Test
Events are dependent
P(A) × P(B) = 0.1200
P(A ∩ B) = 0.1000

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What is Probability Calculator?

Probability calculations help you quantify uncertainty and make informed decisions. This calculator covers basic probability rules, and three important probability distributions used across science, business, and everyday decision-making.

Basic Probability Rules

  • Union (A ∪ B): P(A or B) = P(A) + P(B) - P(A and B)
  • Intersection (A ∩ B): P(A and B) - probability both events occur
  • Conditional (A|B): P(A given B) = P(A ∩ B) / P(B)
  • Independence: P(A ∩ B) = P(A) × P(B) if events don't influence each other
  • Complement: P(not A) = 1 - P(A)

Binomial Distribution

  • Use when: Fixed number of independent trials, each with same success probability
  • Examples: Coin flips, quality control, clinical trials, survey responses
  • Parameters: n (trials), p (success probability per trial)
  • Properties: Mean = np, Variance = np(1-p)

Normal Distribution

  • Use when: Continuous data that follows bell-shaped pattern
  • Examples: Heights, test scores, measurement errors, natural phenomena
  • Parameters: μ (mean), σ (standard deviation)
  • Properties: Symmetric, 68-95-99.7 rule applies

Poisson Distribution

  • Use when: Counting rare events in fixed time/space intervals
  • Examples: Phone calls per hour, accidents per day, defects per batch
  • Parameter: λ (average rate of occurrence)
  • Properties: Mean = Variance = λ

Real-World Applications

  • Business: Risk assessment, quality control, customer behavior prediction
  • Healthcare: Treatment success rates, epidemic modeling, diagnostic accuracy
  • Finance: Investment risk, insurance pricing, market volatility
  • Engineering: Reliability analysis, failure rates, system performance
  • Gaming: Odds calculation, expected outcomes, fair game design

Choosing the Right Distribution

  • Binomial: Fixed trials, binary outcomes (success/failure)
  • Normal: Continuous data, bell-shaped distribution
  • Poisson: Rare events, counting in intervals
  • Consider: Data type, assumptions, real-world context


FAQ - Probability Calculator

Two events are independent if the occurrence of one doesn't affect the probability of the other. Mathematically, P(A ∩ B) = P(A) × P(B). For example, coin flips are independent - getting heads on the first flip doesn't change the probability of heads on the second flip.