Probability
Binomial Event Probability Calculator
Calculate exact observed-count probability, cumulative and threshold probability, mean, variance, standard deviation, expected value, and a comparison mean for a binomial experiment.
Expected successes n × p-
Binomial variance-
Binomial standard deviation-
Probability of exactly observed k-
Probability of X ≤ observed k-
Probability of X ≥ threshold r-
Expected gross value-
Expected net value-
Expected successes at comparison p-
Decision view
Binomial probability mass and threshold tail
Binomial probability mass and threshold tailExact probability bars cover every feasible success count; the observed count, expected count, threshold tail, and comparison mean are marked without substituting a continuous approximation.
Exact scenario comparisonSuccess probability per trial p (%) changes while all other entered assumptions remain constant.
| Success probability per trial p (%) | Expected successes n × p | Binomial variance | Binomial standard deviation | Probability of exactly observed k | Probability of X ≤ observed k | Probability of X ≥ threshold r | Expected gross value | Expected net value | Expected successes at comparison p |
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How to use Binomial Event Probability Calculator
- Enter the fixed trial count and per-trial success probability.
- Choose an observed count and an at-least threshold.
- Add value and comparison assumptions to interpret the distribution.
Calculator guide
Understanding Binomial Event Probability Calculator
A binomial model describes a fixed number of independent yes/no trials with one constant success probability. This page calculates exact masses and tails rather than replacing the discrete distribution with a normal curve.
Detailed calculation process
Detailed exact binomial calculation
The default experiment has 20 trials, 30% success probability, six observed successes, and an eight-success threshold.
General formula: P(X=k)=C(n,k)p^k(1-p)^(n-k)mu=npsigma=sqrt(np(1-p))P(X>=r)=1-sum[P(X=x),x=0..r-1] The combination term counts success placements, the power terms price each placement, and exact masses are summed for a tail.
What each symbol means
Worked substitution with the default inputs
All exact masses from zero through twenty sum to one, and the expected count of six reconciles to the displayed $720 gross value.
Worked situations
Practical examples
- With n=20 and p=30%, the expected count is 6 and the probability of exactly 6 successes is about 19.16%.
- The probability of at least 8 successes sums the exact masses from 8 through 20.
Better inputs
Useful tips
- Confirm every trial has the same success definition.
- Use a probability estimated from comparable trials.
- Inspect the full distribution when the threshold is near the mean.
Before relying on the result
Limitations and common mistakes
- Trials must be independent with constant probability.
- Overdispersion, learning, depletion, and clustered trials violate the model.
- Expected monetary value does not describe outcome variability or utility.
Reference
Key terms
- Probability mass
- Probability assigned to one integer success count.
- Cumulative probability
- Sum of probability masses through a stated count.
- Threshold tail
- Probability assigned to counts at or above the entered threshold.
Important note
A precise binomial answer is only as valid as the fixed-n, constant-p, independence assumptions.
Frequently asked questions
Why is the probability of exactly the mean not close to 100%?
Probability is spread across many feasible integer counts.
Can the observed count exceed n?
No. Keep observed and threshold counts between zero and the trial count.
When should I use Poisson instead?
Poisson can approximate rare events across exposure when a fixed-trial Bernoulli structure is not the natural model.