Binomial Distribution Formula Calculator
Calculate exact and cumulative binomial probabilities with the formula P(X = k) = C(n, k) pk(1-p)n-k.
What Is a Binomial Distribution?
A binomial distribution models the number of successes in a fixed number of independent trials, where each trial has only two outcomes (success or failure) and a constant probability of success. If you flip a coin 10 times and count heads, that count follows a binomial model.
This calculator is useful for common probability questions like:
- What is the chance of getting exactly 4 correct answers out of 8 multiple-choice questions?
- What is the probability of at least 2 defects in a batch sample?
- What is the chance of no failures in 12 independent tests?
Binomial Distribution Formula
Where:
- n = total number of trials
- k = number of successes
- p = probability of success on one trial
- C(n, k) = number of combinations, computed as n! / (k!(n-k)!)
Conditions for Using the Binomial Formula
- The number of trials is fixed in advance.
- Each trial is independent.
- Each trial has exactly two possible outcomes.
- The success probability p stays the same each time.
How to Use This Binomial Probability Calculator
- Enter the number of trials n.
- Enter the success probability p (from 0 to 1).
- Choose exact, at most, at least, or range probability type.
- Enter the required success value(s) and click Calculate Probability.
In addition to probability, the tool also shows the expected value (mean), variance, and standard deviation so you can quickly interpret spread and central tendency.
Worked Example
Suppose a player has a 70% chance of making a free throw, and takes 8 shots. Here, n = 8, p = 0.7. If you want the probability of exactly 6 made shots, compute P(X = 6).
With this calculator, choose Exact, set k = 6, and calculate. You can then switch to At least for P(X ≥ 6) to include outcomes of 6, 7, and 8 makes.
Common Mistakes to Avoid
- Entering percentage as 70 instead of decimal 0.70.
- Using non-integer values for n or k.
- Using a binomial model when trials are not independent.
- Forgetting that cumulative probability includes multiple values of k.
Practical Applications
Binomial probability appears in many real-world contexts:
- Quality control: probability of defective items in a sample.
- Medicine: number of patients responding to treatment.
- Finance: default/no-default event counts across a portfolio segment.
- Marketing: conversions from a fixed number of ad clicks.
- Education: number of correct answers on true/false tests.
Quick FAQ
What if k is greater than n?
The exact probability is zero because you cannot have more successes than trials.
Can this calculator compute cumulative probabilities?
Yes. Use the At most, At least, or Range options.
When should I use a normal approximation instead?
For very large n, a normal approximation can be faster analytically, but this calculator directly computes binomial probabilities up to practical limits.