Calculate probabilities, mean, and variance for the number of successes in independent trials.
Last updated: April 2026
Max 170 trials
Probability per trial (0–1)
Value between 0 and n
Mean (μ)
5.00
Variance (σ²)
2.50
Std Dev (σ)
1.58
P(X = 5)
24.61%
P(X ≤ 5)
62.30%
P(X > 5)
37.70%
Formula: P(X = k) = C(n, k) × p^k × (1 - p)^(n - k)
| Scenario | n | p | Mean (μ) | σ |
|---|---|---|---|---|
| Fair coin flips | 10 | 0.50 | 5.00 | 1.58 |
| Biased coin (heads favored) | 20 | 0.70 | 14.00 | 2.05 |
| Rare event (defect rate) | 100 | 0.02 | 2.00 | 1.40 |
| Survey response (typical) | 1000 | 0.40 | 400.00 | 15.49 |
Mean = n×p. Standard deviation σ = √(n×p×(1-p)). Larger n tends to produce distributions closer to normal shape.
The binomial distribution describes the number of successes in a fixed number of independent trials, where each trial has the same probability of success. It's one of the most important discrete probability distributions and appears frequently in real-world applications.
Key Requirements: Fixed number of trials (n), independent trials, two outcomes per trial (success/failure), constant success probability (p).
Scenario: Flip a fair coin 10 times. What's the probability of getting exactly 5 heads?
Parameters: n = 10 trials, p = 0.5 (fair coin), k = 5 successes
About 1 in 4 sequences of 10 coin flips will yield exactly 5 heads. This probability is highest near the mean (5).
What does 'independent trials' mean?
Each trial's outcome doesn't affect other trials. Flipping a coin is independent; drawing cards without replacement is not (probabilities change).
What's the difference between PMF and CDF?
PMF (P(X = k)) is the probability of exactly k successes. CDF (P(X ≤ k)) is cumulative probability from 0 up to k.
When should I use binomial?
Any fixed number of independent binary trials with constant success probability: quality control, survey responses, disease testing, sports records.
How does changing n or p affect the distribution?
Increasing n makes it wider. Increasing p shifts center right; decreasing p shifts it left. Large n makes it bell-shaped.
Relationship to normal distribution?
For large n, binomial approximates normal with mean np and variance np(1-p). This is the Central Limit Theorem in action.
Can p be greater than 1?
No, p must be between 0 and 1. It represents the chance of success on a single trial.
What if trials aren't independent?
Use hypergeometric distribution instead. Drawing without replacement from finite populations violates the independence assumption.
Binomial vs. Poisson?
Binomial: fixed n, two outcomes. Poisson: models rare events in fixed time/space with unknown count. Use Poisson when n is large and p is small.
Related Tools
Gaussian probabilities.
Z-score from probability.
Rare event frequency.
Time between events.
Equal probability distribution.
Calculate beta probabilities.