Binomial CDF Calculator
A binomial CDF calculator evaluates cumulative probabilities for a binomial random variable. If \(X\) counts successes in \(n\) independent trials with success probability \(p\), then \(X \sim \text{Bin}(n,p)\). The cumulative distribution function (CDF) at an integer \(k\) is the probability of obtaining at most \(k\) successes.
Core Formula Used by a Binomial CDF Calculator
The binomial probability mass function (pmf) is:
The CDF is a cumulative sum of pmf terms:
A binomial CDF calculator is effectively an efficient way to compute the above sum (the same quantity that a binomial probability table provides), especially for large \(n\) where manual addition is impractical.
How to Interpret Common Calculator Options
- Left tail: \(P(X \le k)\) (at most \(k\) successes).
- Strictly less than: \(P(X < k)=P(X \le k-1)\) for integer-valued \(X\).
- Right tail: \(P(X \ge k)=1-P(X \le k-1)\).
- At least one: \(P(X \ge 1)=1-P(X=0)\).
- Between two values: \(P(a \le X \le b)=P(X \le b)-P(X \le a-1)\).
Worked Example (Typical Binomial CDF Calculator Output)
Example inputs: \(n=10\), \(p=0.3\), and \(k=4\). The goal is to compute \(P(X \le 4)\).
The calculation is:
| \(x\) | \(P(X=x)\) | Cumulative \(P(X \le x)\) |
|---|---|---|
| 0 | 0.0282475 | 0.0282475 |
| 1 | 0.1210608 | 0.1493083 |
| 2 | 0.2334744 | 0.3827828 |
| 3 | 0.2668279 | 0.6496107 |
| 4 | 0.2001209 | 0.8497317 |
Therefore, a binomial CDF calculator should report \(P(X \le 4) \approx 0.8497\) for these inputs (rounding depending on settings).
Visualization: PMF Bars With the CDF Shaded Up to \(k\)
Common Pitfalls Checked Before Trusting the Output
- Correct parameter meaning: \(n\) is the number of trials, \(p\) is the success probability per trial, and \(k\) is a success count.
- Inclusive vs exclusive: \(P(X \le k)\) differs from \(P(X < k)\); for integers, \(P(X < k)=P(X \le k-1)\).
- Right tail conversion: \(P(X \ge k)\) is not \(1-P(X \le k)\); the correct identity is \(P(X \ge k)=1-P(X \le k-1)\).
- Range check: \(k\) must satisfy \(0 \le k \le n\); otherwise the probability is 0 or 1 by definition.