Binomial Probability Calculator
Binomial probability calculator
Fast exact or cumulative binomial probability checks.
InputsDistribution4 fieldsLive
Result
P(X = 3) = 0.1171875 (11.71875%)
E[X] = 5, Var(X) = 2.5, SD = 1.581139.
Live update
stats
Advanced options
Distribution review
PMF chart (distribution shape)
- k=00.0977%
- k=10.9766%
- k=24.3945%
- k=311.7188%
- k=420.5078%
- k=524.6094%
- k=620.5078%
- k=711.7188%
- k=84.3945%
- k=90.9766%
- k=100.0977%
PMF and CDF table
P(X = 3)
| k | P(X = k) | P(X <= k) |
|---|---|---|
| 0 | 0.0009765625 | 0.0009765625 |
| 1 | 0.009765625 | 0.0107421875 |
| 2 | 0.0439453125 | 0.0546875 |
| 3 | 0.1171875 | 0.171875 |
| 4 | 0.205078125 | 0.376953125 |
| 5 | 0.24609375 | 0.623046875 |
| 6 | 0.205078125 | 0.828125 |
| 7 | 0.1171875 | 0.9453125 |
| 8 | 0.0439453125 | 0.9892578125 |
| 9 | 0.009765625 | 0.9990234375 |
| 10 | 0.0009765625 | 1 |
Model assumptions
- Trials are independent Bernoulli events.
- Success probability p is constant across all trials.
- k counts whole-number successes only (0 <= k <= n).
- Displayed values are rounded; internal PMF/CDF uses exact binomial terms.
Formula
P(X=k) = C(n k) p^k (1-p)^(n-k) Symbol legend
| Symbol | Meaning | Unit | Copy |
|---|---|---|---|
n | Number of trials | count | |
k | Number of successes | count | |
p | Success probability per trial | 0 to 1 |
- Choose n trials and per-trial success probability p.
- Select probability mode: exact P(X=k), cumulative P(X<=k)/P(X>=k), or range P(a<=X<=b).
- Compute PMF terms and sum over selected k range.
- Review PMF/CDF table and chart to validate how probability mass is distributed.
Example
Worked example: n=10, p=0.5, at-most k=3
- 1 Compute PMF for k=0,1,2,3
- 2 Sum P(X=0)+P(X=1)+P(X=2)+P(X=3)
- 3 Result = 0.171875
The cumulative probability P(X<=3) is 0.171875.
How
- Enter trials n and success probability p.
- Select exact, at-most, at-least, or range mode.
- Enter k (and optional upper bound for range mode).
- Read probability output and use PMF/CDF table plus chart for context.
Avoid
- Using k larger than n.
- Setting p outside zero to one range.
- Applying binomial model to dependent trials.
- Confusing exact P(X=k) with cumulative P(X<=k) or P(X>=k).
FAQ
Do trials need to be independent?
Yes, binomial assumptions require independent trials with constant p.
Can p be zero or one?
Yes, edge cases are allowed and produce deterministic probabilities.
Does this return cumulative probability?
Yes. Choose at-most, at-least, or range mode to sum PMF terms over the selected k interval.
What do PMF and CDF rows mean?
PMF is P(X=k) for one success count, while CDF is P(X<=k) accumulated from zero up to k.
Switch
Switch12
No match.