Probability
Binomial Event Odds Calculator
Calculate exact, cumulative, and upper-tail probabilities for a binomial count, plus its expected value and standard deviation.
Per-trial success probability-
Probability of exactly target successes-
Probability of at most target successes-
Probability below target successes-
Probability of at least target successes-
Expected successes-
Standard deviation of successes-
Decision view
Binomial probability mass with exact and tail emphasis
Binomial probability mass with exact and tail emphasisEvery possible success count has a discrete bar; the target bar and at-least region update with n, p, and k.
Exact scenario comparisonSuccess probability per trial (%) changes while all other entered assumptions remain constant.
| Success probability per trial (%) | Per-trial success probability | Probability of exactly target successes | Probability of at most target successes | Probability below target successes | Probability of at least target successes | Expected successes | Standard deviation of successes |
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How to use Binomial Event Odds Calculator
- Confirm a fixed trial count.
- Use one defensible per-trial probability.
- Choose whether the question means exactly, at most, or at least k.
Calculator guide
Understanding Binomial Event Odds Calculator
A binomial model applies when a fixed number of independent trials share one success probability and each trial has only success or failure.
Detailed calculation process
Detailed binomial event calculation
The default model uses 20 independent trials, 15% success probability, and a target of three successes.
General formula: P(X=k)=C(n,k)p^k(1-p)^(n-k)P(X≤k)=Σ[i=0..k]P(X=i)P(X≥k)=1-P(X≤k-1)E[X]=npσ=√[np(1-p)] The combination term counts placements of successes; probability powers weight each success/failure pattern.
What each symbol means
Worked substitution with the default inputs
The default probability is about 24.28% for exactly three successes and 59.51% for at least three.
Worked situations
Practical examples
- With n=20 and p=15%, the expected count is three.
- Exactly three successes has probability about 24.29%, while at least three is a broader event.
Better inputs
Useful tips
- Model changing probabilities with a different method.
- Check dependence between trials.
- State whether the target outcome is included in a tail.
Before relying on the result
Limitations and common mistakes
- Trials are assumed independent and identically distributed.
- The model does not estimate p from data or add uncertainty around p.
- Counts above 500 are intentionally not supported in this interactive implementation.
Reference
Key terms
- PMF
- Probability mass at one exact discrete count.
- CDF
- Probability accumulated from zero through a selected count.
- Upper tail
- Probability at or above a selected count.
Important note
Validate independence and probability stability before using binomial tails for quality, clinical, financial, or safety decisions.
Frequently asked questions
When is a binomial model inappropriate?
When trials interact, probabilities change materially, or more than two outcomes matter.
Why subtract the CDF at k-1?
That leaves k and every larger count in the at-least event.
Can the expected count be non-integer?
Yes. Expectation is a long-run average, not a possible single outcome.