Probability Calculator

Math

Calculate the probability of single events, two independent or mutually exclusive events (A and B), binomial series distributions, and odds ratios.

Presets:

Joint Probability Metrics

Both A and B: P(A ∩ B)

25.00% (0.2500)

Either A or B: P(A ∪ B)

75.00% (0.7500)

Exactly One: P(A ⊕ B) 50.00%

Neither A nor B 25.00%

Not A: P(A') 50.00%

Not B: P(B') 50.00%

Step-by-Step Probability Derivations

1 1. Probability of Both Events Occurring: P(A ∩ B)

P(A ∩ B) = P(A) × P(B) = 0.5000 × 0.5000 = 0.2500

For independent events, joint probability is the product.

2 2. Probability of Either Event Occurring: P(A ∪ B)

P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 0.5000 + 0.5000 - 0.2500 = 0.7500

Addition rule of probability for combining non-disjoint sets.

3 3. Probability of Exactly One Event: P(A ⊕ B)

P(A ⊕ B) = P(A ∪ B) - P(A ∩ B) = 0.7500 - 0.2500 = 0.5000

Either A happens or B happens, but not both.

4 4. Complements

P(A') = 1 - P(A) = 0.5000, P(Neither) = 1 - P(A ∪ B) = 0.2500

Complementary event probabilities.

How This Calculator Works

Probability quantifies the likelihood that a given event will occur on a normalized scale from 0 (impossible event) to 1 (certain event). It forms the basis of statistical inference, decision theory, risk modeling, and machine learning.

📐 Probability Axioms & Rules

Union: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) | Independent Intersection: P(A ∩ B) = P(A) × P(B)

P(A), P(B): Probability of individual events A and B occurring

P(A ∩ B): Joint probability: both event A and event B occur together

P(A ∪ B): Union: at least one of the events occurs

P(A ⊕ B): Exclusive or: exactly one event occurs, but not both

P(A'): Complement: probability that event A does NOT occur = 1 - P(A)

P(A | B): Conditional probability of A given that B has already occurred

Key Mathematical Rules & Properties

Independent vs Mutually Exclusive

Independent events do not influence one another [P(A ∩ B) = P(A)P(B)]. Mutually exclusive events cannot occur simultaneously [P(A ∩ B) = 0]. Two non-trivial events cannot be both independent and mutually exclusive.

Complement Rule for Series

The probability of an event occurring "at least once" in n independent trials is calculated as 1 minus the probability that it never occurs: P(At least once) = 1 - (1 - p)^n.

Odds vs Probability

Probability is favorable divided by total outcomes [a / (a + b)]. Odds in favor is favorable divided by unfavorable [a / b].

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Worked Example: Rolling a Die 3 Times

What is the probability of rolling at least one 6 in 3 rolls of a standard fair die?

1. Probability of rolling a 6 on any single roll: p = 1/6.
2. Probability of NOT rolling a 6 on a single roll: 1 - p = 5/6.
3. Probability of never rolling a 6 in 3 independent rolls: (5/6)³ = 125/216 ≈ 0.5787.
4. Complement rule: P(At least one 6) = 1 - 125/216 = 91/216 ≈ 0.4213 (42.13%).

Final Answer: 91/216 ≈ 42.13%

Frequently Asked Questions

Can probability ever exceed 100% or be negative?

No. By Kolmogorov’s probability axioms, any valid probability P(E) must strictly satisfy 0 ≤ P(E) ≤ 1 (0% to 100%).

How do you convert odds to probability?

If odds in favor are a : b, the probability is a / (a + b). For example, 3 : 1 odds against an event implies 1 : 3 in favor, so P = 1 / (1 + 3) = 1/4 = 25%.

Understanding your calculation

Method, example and limitations

How the calculation works

Probability ranges from zero to one. Complement, union and intersection calculations depend on the relationship between events.

Example to check

For independent fair coin tosses, the probability of two heads is 1/2 × 1/2 = 1/4.

Before using the result

Multiply probabilities directly only when the required independence or conditional probabilities are justified. Overlapping events cannot simply be added without subtracting their intersection.

Calculation methodology · Report a result that needs review