Find a binomial quantile (inverse CDF). Enter the number of trials, success probability, and a target probability to solve for the cutoff value k.
What this inverse binomial calculator does
The binomial distribution answers questions about the number of successes in a fixed number of independent trials. The inverse binomial problem goes the other direction: instead of asking “what is the probability of getting exactly x successes?”, you ask “what value of x gives me a desired cumulative probability?”
In this calculator, that value is called k. You can solve two useful inverse problems:
- Lower-tail quantile: smallest k where P(X ≤ k) is at least q.
- Upper-tail quantile: largest k where P(X ≥ k) is at least q.
Quick binomial refresher
Parameters
- n = number of trials
- p = probability of success on each trial
- X = number of successes in n trials
Then we write: X ~ Binomial(n, p).
Probability mass function (PMF)
The PMF gives P(X = x), the probability of exactly x successes:
P(X = x) = C(n, x) · px · (1 - p)n - x, for x = 0, 1, ..., n.
Cumulative distribution function (CDF)
The CDF gives P(X ≤ x), the probability of getting x or fewer successes. The inverse calculator searches that cumulative curve to find the cutoff that meets your target probability q.
How to use this tool
- Enter the number of trials n (for example, 25 email campaigns, 40 shots, 100 coin tosses).
- Enter success probability p for a single trial (between 0 and 1).
- Enter your target probability q (between 0 and 1).
- Choose lower-tail or upper-tail quantile.
- Click Calculate to get k and nearby distribution values.
Example interpretation
Suppose n = 20 and p = 0.5. If you choose lower-tail with q = 0.95, the result k tells you the 95th percentile of the number of successes (smallest k such that there is at least a 95% chance of seeing k or fewer successes).
If you choose upper-tail with the same settings, k becomes a confidence threshold for “at least k successes” events.
Common real-world uses
- Quality control acceptance thresholds in manufacturing
- Reliability and pass/fail testing
- Sales conversion planning and campaign forecasting
- A/B testing sanity checks and probability thresholds
- Risk and compliance trigger levels
Tips and caveats
Independence matters
The binomial model assumes each trial is independent and has the same p. If your process changes over time (seasonality, learning effects, changing traffic quality), the model can be biased.
Discrete outcomes create jumps
Quantiles in discrete distributions are step-like. That means you may not hit q exactly; instead you’ll get the first (or last) integer k that satisfies the chosen inequality.
Extreme p values
When p is very close to 0 or 1, results can be concentrated near 0 or n. That is expected behavior.
Frequently asked questions
Is this the same as normal distribution inverse?
No. This is exact binomial quantile logic for discrete counts, not a continuous normal approximation.
Can I use decimals for n?
No. n must be a whole number because it represents a count of trials.
Why do I get k = 0 for upper-tail with high q?
Because P(X ≥ 0) = 1 always. For upper-tail interpretation, focus on the largest k that still meets your desired probability. That gives the strongest guaranteed threshold.