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🎯 Binomial Distribution Calculator

Calculate P(X = k), P(X ≤ k), and P(X ≥ k) for a binomial distribution. Enter the number of trials (n), probability of success (p), and number of successes (k) to get exact probabilities, the mean, variance, and a full distribution table.

Total number of trials (1–1000)

Probability per trial (0 to 1)

Specific number of successes to query

Binomial Distribution Formulas

Probability Mass Function
P(X = k) = C(n, k) × p^k × (1−p)^(n−k)
Cumulative (at most k)
P(X ≤ k) = Σ P(X=i) for i = 0 to k
Mean, Variance, SD
μ = n × p
σ² = n × p × (1 − p)
σ = √(n × p × (1 − p))

C(n, k) is the number of ways to choose k successes from n trials. The binomial distribution models the number of successes in n independent trials, each with a constant success probability p.

How to Use This Calculator

  1. 1
    Enter n (trials)
    The total number of independent trials or experiments (e.g. 10 coin flips, 20 free throws).
  2. 2
    Enter p (probability)
    The probability of success on each single trial (0 to 1). For a fair coin, p = 0.5.
  3. 3
    Enter k (successes)
    The specific number of successes you want to query. The calculator gives P(X=k), P(X≤k), and P(X≥k).
  4. 4
    Read the table
    The distribution table shows the exact probability for every possible outcome from 0 to n, with your chosen k highlighted.

Worked Example

A basketball player has a free-throw success rate of p = 0.3. In n = 10 attempts, what is the probability of exactly k = 3 successes?

P(X=3) = C(10,3) × 0.3³ × 0.7⁷
= 120 × 0.027 × 0.0824
= 0.2668 (≈ 26.68%)
P(X≤3) = 0.6496
P(X≥3) = 0.6172
μ = 10 × 0.3 = 3.0, σ = √(10×0.3×0.7) = 1.449
'When should I use the binomial distribution?', 'answer' => 'Use it when: (1) there is a fixed number of trials n, (2) each trial has exactly two outcomes (success/failure), (3) the probability p is constant for every trial, and (4) trials are independent of each other. These are the BINS conditions.'], ['question' => 'Binomial vs normal distribution?', 'answer' => 'The binomial is exact but discrete. The normal distribution is a continuous approximation that works well when both np ≥ 5 and n(1−p) ≥ 5. For small n or extreme p values, always use the exact binomial formula. For large n with moderate p, the normal approximation is accurate.'], ['question' => 'What is the expected value (mean)?', 'answer' => 'The expected value μ = n × p is the average number of successes you would expect over many repetitions of the experiment. For 10 coin flips with p = 0.5, you expect μ = 5 heads on average, though any single experiment may give a different result.'], ]" />

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