Probability Calculator (Single & Multiple Events)

Calculate the probability of simple or complex events. Our tool handles independent and dependent events, unions, intersections, and even conditional probability.

Probability Calculator

Calculate probabilities for common statistical distributions

Distribution

Calculation Type

Mean (μ)

Standard Deviation (σ)

X Value

About Probability Distributions

Normal: Continuous distribution for naturally occurring phenomena. Defined by mean (μ) and standard deviation (σ).

Binomial: Discrete distribution for number of successes in n trials. Defined by trials (n) and success probability (p).

Poisson: Discrete distribution for count of events in fixed interval. Defined by rate (λ).

Exponential: Continuous distribution for time between events. Defined by rate (λ).

How the Probability Calculator Works

Enter probabilities as decimals between 0 and 1, or as percentages. For single events, input the probability directly. For multiple events, enter each event's probability and specify their relationship.

Select the calculation type: union (A or B), intersection (A and B), conditional probability (A given B), or complement (not A). For independent events, the calculator uses P(A and B) = P(A) × P(B). For dependent events, use conditional probability formulas.

Results display as both decimal and percentage. Step-by-step calculations show the formula used. Visual representations help understand the relationship between events using Venn diagrams or probability trees.

When You'd Actually Use This

Risk assessment calculations

Calculate probability of system failure when multiple components can fail. Use union for "any component fails" or intersection for "all components fail."

Medical test interpretation

Find probability of disease given a positive test result. Use conditional probability with test sensitivity, specificity, and disease prevalence.

Game strategy decisions

Calculate odds of drawing specific cards or rolling certain combinations. Compare probabilities to make optimal gameplay decisions.

Quality control sampling

Determine probability of finding defects in a sample. Use binomial probability when sampling with replacement or from large populations.

Investment portfolio analysis

Calculate probability of multiple investments succeeding. Account for correlation between assets when events aren't independent.

Weather forecast planning

Find probability of rain on at least one day of a trip. Combine daily forecasts using union probability for multi-day event planning.

What to Know Before Using

Probabilities must be between 0 and 1.Zero means impossible, one means certain. Percentages need conversion: 50% = 0.5. Values outside this range indicate input errors.

Independence affects calculations.Independent events don't influence each other's probability. Dependent events require conditional probability. Don't assume independence without justification.

Mutually exclusive events can't both occur.For mutually exclusive events, P(A and B) = 0. The union simplifies to P(A) + P(B) with no overlap to subtract.

Conditional probability reverses with Bayes' theorem.P(A|B) differs from P(B|A). Bayes' theorem connects them: P(A|B) = P(B|A) × P(A) / P(B). Crucial for medical testing and inference.

Pro tip: For "at least one" problems, calculate the complement (none occur) and subtract from 1. Example: P(at least one head in 3 flips) = 1 - P(no heads) = 1 - 0.125 = 0.875.

Common Questions

What's the difference between union and intersection?

Union (A or B) means either event happens. Intersection (A and B) means both happen together. Union probability is always greater than or equal to intersection.

How do I know if events are independent?

Events are independent if one doesn't affect the other's probability. Coin flips are independent. Drawing cards without replacement is dependent.

Can probability be greater than 1?

No. Probability ranges from 0 to 1 (or 0% to 100%). Values outside this range indicate calculation errors or misunderstanding of the problem.

What does conditional probability mean?

P(A|B) is the probability of A given that B already occurred. It updates your assessment based on new information. Foundation of Bayesian reasoning.

Why subtract P(A and B) in union formula?

P(A) + P(B) counts the overlap twice. Subtracting P(A and B) removes the double-counting. This is the inclusion-exclusion principle.

How do I handle three or more events?

Apply formulas iteratively. For union of three events: P(A or B or C) = P(A) + P(B) + P(C) - P(A and B) - P(A and C) - P(B and C) + P(A and B and C).

What's the complement rule?

P(not A) = 1 - P(A). Useful for "at least one" problems. Sometimes easier to calculate the probability of something not happening.

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Probability Calculator (Single & Multiple Events)