🎓 Statistics & Data • Probability & Distributions
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Binomial Distribution Calculator (B(n, p))
Computes binomial probability P(X = k) = C(n, k)·p^k·(1-p)^(n-k) with mean and variance.
Discrete Binomial Distribution
Binomial Probability: B(n, p)
Computes exact P(X = k), cumulative P(X ≤ k), expected value μ = np, and variance for n independent Bernoulli trials.
Presets:
Exact Probability P(X = 6)
20.51%
Exact Decimal = 0.2051
P(X ≤ 6)82.81%
P(X ≥ 6)37.7%
Expected μ = np5
Std Dev σ1.5811
Binomial PMF Probability Mass Function Bars
Active k = 6 Cumulative x ≤ 6 Other k values
What is the Binomial Distribution?
The binomial distribution gives the probability of obtaining exactly k successes in n independent Bernoulli trials, each having success probability p.
Formula & Step-by-Step Calculation
P(X = k) = (n! / [k!(n - k)!]) · p^k · (1 - p)^(n - k)
n = trials, k = successes, p = probability of success.
Worked Step-by-Step Examples
Example 1
Toss a fair coin (p=0.5) 10 times. Probability of exactly 6 heads?
Solution: P(X = 6) = 20.51%
• C(10, 6) × (0.5)⁶ × (0.5)⁴ = 210 × 0.0009765 = 0.2051
Common Real-World & Academic Use Cases
- ✓ Quality inspection batch sampling defect rates
- ✓ Clinical drug treatment cure rates
- ✓ Telecommunication packet loss modeling
How to Use the Binomial Distribution Calculator (B(n, p))
1
Enter Number of Trials n
Input sample size.
2
Enter Success Probability p
Input decimal (0 to 1).
3
Enter Successes k
Read exact probability.
Frequently Asked Questions
Q: When does a binomial distribution approximate a normal distribution?
When both np ≥ 10 and n(1 - p) ≥ 10.