Compare P(X = k) and P(X ≤ k)?

P(X = k): exact. P(X ≤ k): cumulative (k or fewer). Use ≤ for one-tailed hypothesis tests.

Normal approximation?

Valid when np ≥ 5 and n(1-p) ≥ 5. Use Z = (X − np) / √(np(1-p)).

Single trial (n = 1). Bernoulli(p) is the building block of Binomial(n, p).

Sampling without replacement (e.g. card draws). Different formula. Reduces to binomial for large populations.

Binomial Probability Calculator

P(X = k) = C(n,k) p^k (1−p)^(n−k). Probability of exactly k successes in n trials.

Result

P(X = k)

0.266828

C(10,3) × 0.3^3 × 0.7000^7.

n10
k3
p0.3
C(n, k)120
P(X = k)0.26682793
P(X ≤ k)0.64961072
P(X > k)0.35038928
E[X] = np3.0000
Var[X] = np(1-p)2.1000

Step-by-step

C(10, 3) = 120.
p^k = 0.3^3 = 0.027000.
(1−p)^(n−k) = 0.7000^7 = 0.082354.
P = 120 × 0.027000 × 0.082354 = 0.266828.

How to use this calculator

Enter n trials, k successes, p success probability.
Read P(X = k), cumulative, and moments.

About this calculator

Binomial distribution models the count of successes in n independent trials, each with probability p. Mean = np; variance = np(1−p). For large n, approximated by normal N(np, np(1−p)). Examples: number of heads in 10 coin flips (n=10, p=0.5), number of defective items, number of yes-responses in a survey. Foundational for hypothesis testing of proportions.