Point versus tail
P(X=8) answers one-count likelihood; P(X>=8) answers an exceedance question. Substituting one for the other changes the decision.
Probability
Binomial Event Distribution Calculator
Build the complete exact binomial distribution for n independent equal-probability trials and inspect one count, lower tail, upper tail, center, spread, and mode.
EXACT DISCRETE DISTRIBUTION
Inspect every possible success count before choosing a tail
The page enumerates k=0 through n with exact binomial mass, cumulative probability, inclusive upper tail, and standardized distance from the mean.
P(X = selected k) -
P(X <= selected k) -
P(X >= selected k) -
Mean success count -
Standard deviation -
Modal count -
LIVE DECISION RECORD
Complete binomial probability table
The exact rows sum to 100%; both inclusive tails retain the selected count.
| Successes k | P(X=k) | P(X<=k) | P(X>=k) | Distance from mean (z) |
|---|
CURRENT CALCULATION PROCESS
Formula, current substitution, intermediate values, and reconciliation
P(X=k)=C(n,k)p^k(1-p)^(n-k); mu=np; sigma^2=np(1-p)
| Symbol | Meaning and unit | Current value |
|---|---|---|
| trials | Independent trial count - Whole Bernoulli trials in one experiment. | 20 |
| successProbabilityPct | Success probability per trial (%) - Stable probability assigned to every trial. | 35 |
| selectedSuccesses | Selected success count - Whole count k for exact and cumulative queries. | 8 |
Waiting for valid inputs.
WHO THIS MODEL SERVES
A scoped decision aid, not a universal forecast
Primary audience: Analysts, students, quality teams, and planners who need an exact finite-trial count distribution.
Decision boundary: Use only when trial count is fixed, trials are independent, and one success probability applies to every trial.
HOW TO READ THE DISTRIBUTION
Five steps from trial definition to tail choice
- Define one trial and the event counted as success.
- Confirm n is fixed before outcomes are observed.
- Enter a stable per-trial probability on the same population basis.
- Select k and distinguish exact, lower-tail, and inclusive upper-tail questions.
- Verify row closure and export the table with the event definition.
BINOMIAL FUNDAMENTALS
Five properties behind the rows
- Fixed n
- The experiment contains a predetermined whole number of trials.
- Binary outcome
- Each trial is classified as success or not-success.
- Constant p
- Every trial uses the same event probability.
- Independence
- One result does not alter another trial's probability.
- Mutually exclusive counts
- Exactly one total k occurs, so all row masses sum to one.
FORMULA AND DEFAULT SUBSTITUTION
Keep combinations separate from sequence order
P(X=k)=C(n,k)p^k(1-p)^(n-k)
At n=20, p=0.35, and k=8, the exact row is C(20,8) x 0.35^8 x 0.65^12. The mean is 20 x 0.35=7 and variance is 20 x 0.35 x 0.65=4.55; live rows remain unrounded until display.
DEEPER DISTRIBUTION ANALYSIS
Three ways to interrogate the shape
Center and skew
The mean need not be a possible integer outcome, and distributions near p=0 or p=1 become strongly asymmetric.
Inclusive boundaries
The selected row belongs to both displayed tails; complement checks must shift the boundary by one count.
WORKED DISTRIBUTION CASES
Two count questions with different table reads
Twenty inspections
If "success" means a defect and p=35%, the expected count is seven. Selecting eight reveals the chance of exactly eight defects and the chance of eight or more - different escalation signals.
Deterministic boundary
At p=0, the k=0 row carries 100% and every other row carries zero. This verifies that the model handles degenerate probabilities without undefined spread.
DISTRIBUTION TERMINOLOGY
Six terms used in the table
- Combination
- Number of unordered ways k successes can appear among n trials.
- PMF
- Probability mass assigned to one exact count.
- CDF
- Accumulated probability through a selected count.
- Upper tail
- Probability at or above the selected count.
- Variance
- np(1-p), the count distribution's squared spread.
- Mode
- A count with greatest probability mass.
EVIDENCE RETENTION
Preserve the trial protocol behind n and p
Record population, sampling frame, success rule, exclusions, probability source, dependence review, and whether trials were fixed in advance. Keep the exported table with the selected tail question.
LIMITS AND EXCLUSIONS
When exact binomial rows are not enough
- Trials require a stable p and credible independence.
- Changing exposure or unequal probabilities need another model.
- The table describes process variability, not uncertainty in the estimated p.
- Selection bias and classification error are outside the formula.
- Decisions with dependence or overdispersion need a reviewed alternative.
RELIABLE SOURCES
Primary references for the exact distribution
EXACT-DISTRIBUTION FAQ
Questions about masses, tails, and boundaries
What makes an event binomial?
A fixed number of independent trials, two outcomes per trial, and one stable success probability.
Why are P(X<=k) and P(X>=k) not complements?
Both include the selected row k. The complement of P(X<=k) is P(X>=k+1).
Can success mean an undesirable event?
Yes. Success is merely the counted outcome; label it explicitly in the evidence record.
What happens at p=0 or p=1?
All probability collapses to k=0 or k=n respectively, which is a valid degenerate distribution.
Why show a z-position?
It expresses each row's distance from the mean in standard-deviation units; it is descriptive, not a normal approximation.
How many trials can this page enumerate?
Up to 500, producing 501 exact rows; larger problems need a reviewed numerical workflow.
IMPORTANT MODEL NOTE
Exact arithmetic does not rescue a wrong trial model
The probabilities are exact for the entered binomial assumptions. Validate trial definition, p stability, and independence before interpreting small tail values.