Binomial CDF Calculator
Calculate cumulative distribution function probabilities for binomial distributions
Input Parameters
Results
Binomial Distribution Formulas
P(X = k) = C(n,k) × p^k × (1-p)^(n-k)
P(X ≤ k) = Σ(i=0 to k) C(n,i) × p^i × (1-p)^(n-i)
Variance: σ² = n × p × (1-p)
Standard Deviation: σ = √(n × p × (1-p))
What is Binomial Distribution?
The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success.
It requires four conditions: fixed number of trials, independent trials, two possible outcomes per trial, and constant probability of success.
Cumulative Distribution Function
The CDF gives the probability that a random variable takes a value less than or equal to a specific value.
For binomial distributions, it calculates the probability of getting k or fewer successes in n trials.
Applications
Quality control in manufacturing, medical testing accuracy, survey analysis, and any scenario with binary outcomes.
Common examples include coin flips, pass/fail tests, and yes/no survey responses.
Key Properties
The distribution is symmetric when p = 0.5, right-skewed when p < 0.5, and left-skewed when p > 0.5.
As n increases, the binomial distribution approaches a normal distribution under certain conditions.
Practical Examples
Solution: n = 10, p = 0.5, x = 6
P(X = 6) = C(10,6) × 0.5^6 × 0.5^4 = 210 × 0.015625 × 0.0625 = 0.205
Solution: n = 20, p = 0.05, calculate P(X ≤ 2)
This requires summing P(X = 0) + P(X = 1) + P(X = 2) = 0.925
Solution: n = 15, p = 0.3, calculate P(3 ≤ X ≤ 7)
This equals P(X ≤ 7) – P(X ≤ 2) = 0.950 – 0.127 = 0.823