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Binomial Distribution

Created:August 11, 2024
Last Updated:March 26, 2025

This calculator helps you compute the probabilities of a binomial distribution. Simply enter the number of trials (n), the probability of success (p), and the desired comparison type and value. The calculator will display the probability distribution chart and statistics.

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Binomial Distribution: Definition, Formula, and Examples

Binomial Distribution

Definition: The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials. A Bernoulli trial is an experiment with two possible outcomes: success or failure.

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

Where:

  • nn is the number of trials
  • kk is the number of successes
  • pp is the probability of success on each trial
  • \inom{n}{k} is the binomial coefficient
Examples: Suppose you flip a fair coin 1010 times (n=10,p=0.5)(n = 10, p = 0.5). Let's calculate various probabilities:
  1. Probability of getting exactly 6 heads:P(X=6)=(106)(0.5)6(0.5)10−6≈0.2051P(X = 6) = \binom{10}{6} (0.5)^6 (0.5)^{10-6} \approx 0.2051
  2. Probability of getting between 3 and 7 heads (3<X<7)(3 < X < 7):P(3<X<7)=P(X=4)+P(X=5)+P(X=6)≈0.6563P(3 < X < 7) = P(X = 4) + P(X = 5) + P(X = 6) \approx 0.6563
  3. Probability of getting less than 5 heads (X<5)(X < 5):P(X<5)=P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)≈0.3770P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) \approx 0.3770
  4. Probability of getting more than 4 heads (X>4)(X > 4):P(X>4)=1−P(X≤4)≈0.6230P(X > 4) = 1 - P(X \leq 4) \approx 0.6230

Properties of Binomial Distribution

  • Mean: E(X)=npE(X) = np
  • Variance: Var(X)=np(1−p)Var(X) = np(1-p)
  • Standard Deviation: SD(X)=np(1−p)SD(X) = \sqrt{np(1 - p)}

How to Calculate Binomial Probabilities in R?

R
library(tidyverse)
n <- 10
p <- 0.5

# P(X = 6)
# dbinom(k, size, prob) returns the probability of getting k successes in n trials
prob_equal_6 <- dbinom(6, size = n, prob = p)
print(prob_equal_6) # 0.205078125

# P(X <= 4)
# pbinom(k, size, prob) returns the probability of getting at most k successes in n trials
prob_less_equal_4 <- pbinom(4, size = n, prob = p)
print(prob_less_equal_4) # 0.376953125

# P(X > 7)
# P(X > 7) = 1 - P(X <= 7)
prob_greater_7 <- 1 - pbinom(7, size = n, prob = p)
print(prob_greater_7) # 0.0546875

# P(3 < X < 8)
# P(3 < X < 8) = P(X <= 7) - P(X <= 3)
prob_between_3_and_8 <- pbinom(7, size = n, prob = p) - pbinom(3, size = n, prob = p)
print(prob_between_3_and_8) # 0.7734375

# mean and variance
print(str_glue("Mean: {n * p}")) # 5
print(str_glue("Variance: {n * p * (1 - p)}") # 2.5

How to Calculate Binomial Probabilities in Python?

Python
import scipy.stats as stats

n = 10
p = 0.5

# P(X = 6)
# stats.binom.pmf(k, n, p) returns the probability of getting k successes in n trials
prob_equal_6 = stats.binom.pmf(6, n, p)
print(prob_equal_6)

# P(X <= 4)
# stats.binom.cdf(k, n, p) returns the probability of getting at most k successes in n trials
prob_less_equal_4 = stats.binom.cdf(4, n, p)
print(prob_less_equal_4)

# P(X > 7)
prob_greater_7 = 1 - stats.binom.cdf(7, n, p)
print(prob_greater_7)

# P(3 < X < 8)
prob_between_3_and_8 = stats.binom.cdf(7, n, p) - stats.binom.cdf(3, n, p)
print(prob_between_3_and_8)

# mean and variance
print(f"Mean: {n * p}")
print(f"Variance: {n * p * (1 - p)}")

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