Skip to content

Probability Formulas for Events and Binomial Trials

Review complement, intersection, union, conditional, and binomial probability formulas with variable definitions and calculation steps.

How do the Probability Calculator formulas work?

This formula page defines the variables used by the Probability Calculator, shows how those values are combined, and explains the order of operations behind the result. Use the worked steps to verify an estimate, compare scenarios, and understand which assumptions or rounding rules can affect the displayed answer.

Formula Overview

Event formulas combine sets of outcomes. Binomial formulas describe the number of successes in repeated independent trials with a constant success probability.

Variable Definitions

  • P(A) and P(B) are event probabilities.
  • P(A ∩ B) is the intersection.
  • P(A ∪ B) is the union.
  • n is the number of trials.
  • k is the target success count.
  • p is success probability per trial.

Complement Formula

P(Ac)=1−P(A)P(A^c)=1-P(A)

The complement contains every outcome outside event A.

Union Formula

P(A∪B)=P(A)+P(B)−P(A∩B)P(A\cup B)=P(A)+P(B)-P(A\cap B)

The intersection is subtracted because it appears in both individual probabilities.

Independent Intersection

P(A∩B)=P(A)P(B)P(A\cap B)=P(A)P(B)

This multiplication rule applies only when A and B are independent.

Conditional Probability

P(A∣B)=P(A∩B)P(B),P(B)>0P(A\mid B)=\frac{P(A\cap B)}{P(B)},\quad P(B)>0

Binomial Probability

P(X=k)=(nk)pk(1−p)n−kP(X=k)=\binom{n}{k}p^k(1-p)^{n-k}

The binomial coefficient counts the arrangements containing exactly k successes.

Cumulative Binomial Probability

P(X≤k)=∑i=0k(ni)pi(1−p)n−iP(X\le k)=\sum_{i=0}^{k}\binom{n}{i}p^i(1-p)^{n-i}
P(X≥k)=∑i=kn(ni)pi(1−p)n−iP(X\ge k)=\sum_{i=k}^{n}\binom{n}{i}p^i(1-p)^{n-i}

Mean and Spread

μ=np,σ2=np(1−p),σ=np(1−p)\mu=np,\qquad \sigma^2=np(1-p),\qquad \sigma=\sqrt{np(1-p)}

Displayed values are rounded to two or three decimal places only after calculations are complete.

Worked Calculation

For independent events with P(A) = 0.40 and P(B) = 0.30, the intersection is 0.40 × 0.30 = 0.12. The union is 0.40 + 0.30 - 0.12 = 0.58, so neither occurs with probability 1 - 0.58 = 0.42.

Formula Assumptions

  • Every probability is between 0 and 1.
  • Multiplication for intersections requires independence.
  • Conditional probability requires a nonzero conditioning probability.
  • Binomial calculations require fixed n, independent trials, two outcomes, and constant p.

Next Steps

Apply these formulas in the Probability Calculator or verify them with the worked probability examples.

Use the Probability Calculator