Binomial Probability Calculator
Calculate the probability of observing exactly k successes in n independent Bernoulli trials with constant probability of success p (such as coin flips, quality control defect rates, or clinical recovery rates).
Total independent experiments or attempts
Exact number of successful outcomes desired
Chance of success on any single attempt (between 0.0 and 1.0)
Likelihood of achieving exactly k successes (%)
How to Use This Calculator
- Enter the total number of attempts or trials (n).
- Enter the number of target successes (k).
- Set the single-trial probability (p) between 0 and 1 (e.g. 0.5 for coin flip).
- Inspect the exact probability percentage, expected mean, and variance.
Formula & Calculation Methodology
Multiplies the number of combinations of choosing k successes from n attempts C(n, k) by the probability of k successes (p^k) and (n − k) failures ((1 − p)^(n − k)).
Worked Example
Tossing a fair coin 10 times: Exactly 4 heads (p = 0.5)
Rolling 3 ones in 5 rolls of a 6-sided die (p = 0.1667)
Frequently Asked Questions
What are the assumptions of a binomial distribution?↓
Trials must be fixed in number (n), independent of one another, binary (only success or failure), with a constant probability of success (p) on each trial.
When can a binomial distribution be approximated by a normal distribution?↓
When both np ≥ 5 and n(1 − p) ≥ 5, the binomial distribution becomes sufficiently symmetric to approximate with a Gaussian normal curve.
What occurs if target successes k exceeds total trials n?↓
It is impossible to observe more successes than trials, so the probability is zero.
What is a limitation of the binomial model?↓
It assumes trials are with replacement or drawn from an infinite population. For finite pools without replacement (like drawing cards), use the hypergeometric distribution instead.
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Source Citation & Disclaimer
Informational tool only; not licensed tax, legal, or financial advice.