Probability Calculator

Probability of Two Events

Probability measures the likelihood that an event occurs and is expressed between 0 (impossible) and 1 (certain). The calculator reproduces the calculator.net workflow by letting you evaluate complements, intersections, unions, and XOR results for two independent events.

In the most general case probability equals the number of desired outcomes divided by total outcomes and is affected by whether events are independent, mutually exclusive, or conditional.

Complement of A and B

Given a probability P(A), the complement P(A') describes the chance that A does not occur and equals 1 - P(A). Example: if Bob fails to do homework with P(A) = 0.65, then P(A') = 1 - 0.65 = 0.35, so there is a 35% chance he completes it.

Each event keeps its own complement: P(B) can take any value independent from P(A), and P(B') is computed the same way.

Intersection of A and B

The intersection P(A ∩ B) (also written P(A AND B)) is the joint probability that both events occur. If events are mutually exclusive, the intersection equals zero (rolling a 4 and 6 on the same die roll is impossible).

For independent events, multiply the probabilities: P(A ∩ B) = P(A) × P(B). The bag-of-marbles example from calculator.net shows drawing a blue marble (3/10) then a black marble without replacement, leading to P(A ∩ B) = (3/10) × (7/9) = 0.2333 once conditional probability is considered.

Union of A and B

The union P(A ∪ B) (or P(A OR B)) covers outcomes where either or both events occur. For mutually exclusive events the union is simply the sum of probabilities.

When events are not mutually exclusive, inclusion-exclusion applies: P(A ∪ B) = P(A) + P(B) - P(A ∩ B). Calculator.net illustrates this with a die where event A is rolling an even number and event B is rolling a multiple of three. P(A) = 3/6, P(B) = 2/6, P(A ∩ B) = 1/6, so P(A ∪ B) = 2/3.

Exclusive OR of A and B

P(A XOR B) captures the probability that exactly one event occurs (A or B but not both). The equation becomes P(A) + P(B) - 2 × P(A ∩ B).

Calculator.net's story describes Halloween candy buckets: with P(A) = 0.65 for Reese's, P(B) = 0.349 for Snickers, and a tiny overlap of 0.001, the probability that a child takes exactly one candy equals 0.65 + 0.349 - 2 × 0.65 × 0.349 = 0.5453.

Normal Distribution

The normal (Gaussian) distribution is continuous and follows f(x) = (1/(σ√(2π))) × e^{-((x - μ)^2)/(2σ^2)} where μ is the mean and σ the standard deviation. When μ = 0 and σ = 1 it becomes the standard normal distribution.

The calculator.net walkthrough computes the probability that a male student is between 60 and 72 inches tall when μ = 68 and σ = 4. Standardizing produces z-scores of -2 and 1, and consulting the Z table shows the total probability between those bounds is approximately 0.81859.

Because the normal distribution is symmetric, the area from 0 to 2 equals the area from 0 to -2, allowing the table of positive z-values (0 to Z) to answer both sides. An accompanying confidence-interval table further supports statistical analysis.