Binomial Probability Calculator
Calculate exact and cumulative probabilities for a binomial random variable. Enter your values for trials n, success probability p, and success count(s).
What is a Binomial Variable?
A binomial variable counts how many times a “success” happens in a fixed number of repeated trials. Each trial can only have two outcomes (success or failure), and the probability of success stays the same in every trial.
Examples include:
- How many heads appear in 20 coin flips
- How many customers click an ad out of 100 impressions
- How many parts fail quality testing in a sample of 50
Binomial Distribution Formula
If X ~ Binomial(n, p), then the probability of getting exactly k successes is:
P(X = k) = C(n, k) * p^k * (1 - p)^(n-k)
Where:
- n = total number of trials
- p = probability of success on each trial
- k = number of successes
- C(n, k) = combinations, often read as “n choose k”
Expected Value and Spread
- Mean:
μ = n × p - Variance:
σ² = n × p × (1 - p) - Standard deviation:
σ = √(n × p × (1 - p))
How to Use This Binomial Variable Calculator
- Enter the number of trials n.
- Enter success probability p (between 0 and 1).
- Choose a calculation type:
- P(X = k) for exact probability
- P(X ≤ k) for left-tail cumulative probability
- P(X ≥ k) for right-tail cumulative probability
- P(a ≤ X ≤ b) for interval probability
- Moments for mean/variance/standard deviation only
- Click Calculate.
Worked Example
Suppose a sales email has a 30% response rate. You send it to 12 people. Let X be the number of responses:
n = 12p = 0.30
If you want the chance of getting at least 5 responses, choose P(X ≥ k) and enter k = 5. The calculator gives the cumulative right-tail probability instantly, along with the expected value and spread.
Common Mistakes to Avoid
1) Using percentages instead of decimals for p
Enter 0.25 instead of 25.
2) Forgetting the binomial assumptions
The model requires independent trials and constant probability. If probability changes from trial to trial, a different model may be better.
3) Mixing up exact vs. cumulative probability
P(X = k) is one specific outcome. P(X ≤ k) and P(X ≥ k) add probabilities across many outcomes.
Where This Helps in Real Life
- A/B testing: conversions out of visitors
- Quality control: defect counts in batches
- Healthcare: positive tests in a screened group
- Finance and risk: defaults in a portfolio sample
- Education: correct answers on true/false items
Final Note
This binomial variable calculator is ideal when outcomes are binary and trial conditions are stable. For larger samples, you can also compare with normal approximation methods, but this calculator gives the direct binomial result so you can trust the exact value.