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, as well as the mean and variance of the 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.
Where:
When n is large and p is small , the binomial distribution can be approximated by a Poisson distribution with parameter λ = np:
This approximation is generally considered good when n ≥ 20, p ≤ 0.1, and np ≤ 10.
Example: If you have n = 1000 trials with p = 0.005 probability of success, instead of using the binomial formula, you can approximate using Poisson with λ = 1000 × 0.005 = 5.
When the sample size (n) is large, the binomial distribution can be approximated by a normal distribution with:
This approximation works well when both and .
When using the normal approximation for binomial probabilities, a continuity correction should be applied:
The normal approximation is particularly useful for calculating binomial probabilities when n is large, as exact calculations become computationally intensive.
Adjust n (number of trials) and p (probability of success) to see when the binomial distribution approximates a normal distribution.
Mean (μ): 15.00
Standard Deviation (σ): 2.74
Key Observations:
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)}") # 2.5import 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)}")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, as well as the mean and variance of the 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.
Where:
When n is large and p is small , the binomial distribution can be approximated by a Poisson distribution with parameter λ = np:
This approximation is generally considered good when n ≥ 20, p ≤ 0.1, and np ≤ 10.
Example: If you have n = 1000 trials with p = 0.005 probability of success, instead of using the binomial formula, you can approximate using Poisson with λ = 1000 × 0.005 = 5.
When the sample size (n) is large, the binomial distribution can be approximated by a normal distribution with:
This approximation works well when both and .
When using the normal approximation for binomial probabilities, a continuity correction should be applied:
The normal approximation is particularly useful for calculating binomial probabilities when n is large, as exact calculations become computationally intensive.
Adjust n (number of trials) and p (probability of success) to see when the binomial distribution approximates a normal distribution.
Mean (μ): 15.00
Standard Deviation (σ): 2.74
Key Observations:
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)}") # 2.5import 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)}")