Binomial CDF Calculator
Add up the binomial probabilities for you. Get P(X ≤ k), fewer than, at least, more than or a range for any n and p, or work backwards from a target probability to the cut-off k, with every term of the sum shown.
Cumulative Binomial Probability Calculator
Choose the question, enter n and p, then k. Results update as you type.
n up to 100,000. For “at most k” use ≤. For “fewer than k” use <.
Working appears here.Syntax appears here.Summary appears here.At most, fewer than, at least, more than and between, each with the boundary handled correctly.
Enter a target such as 95% and get the smallest k that reaches it, with the two neighbouring values shown.
The smaller tail is summed directly, so tiny probabilities keep their digits. Checked against SciPy.
What Does a Binomial CDF Calculator Do?
A binomial CDF calculator adds up binomial probabilities for you. Give it the number of trials n, the success probability p and a cut-off k, and it returns P(X ≤ k), the chance of getting k successes or fewer. For 10 fair coin flips and k = 4 the answer is 0.3770.
The same tool answers the other three wordings, “fewer than”, “at least” and “more than”, and the interval question “between a and b”. It also runs backwards. Enter a target probability such as 95% and it finds the smallest k whose cumulative probability reaches it.
Each answer comes with the terms that were added, the complement, the exact probability P(X = k), a table of the running totals and a chart showing the bars and the cumulative curve together. If you came here looking for binomcdf on a TI-84, this is that function with the working shown.
How Do You Use This Binomial CDF Calculator?
Pick the type of cumulative probability, enter n and p, then enter k. Results update as you type.
- Choose the question. At most (≤), fewer than (<), at least (≥), more than (>), between two values, or the inverse mode that finds k from a probability.
- Enter n and p. n is a whole number of trials. p can be a decimal such as 0.25, a fraction such as 1/6, or a percentage such as 25%.
- Enter k (or a and b for a range, or the target probability for the inverse mode).
- Read the result cards. The main card shows the probability, and the other cards show the percentage, the complement and P(X = k).
Open the working panel to see which terms were summed. Open the table to see P(X = k), P(X ≤ k) and P(X ≥ k) side by side for every k near your answer. The inputs accept n up to 100,000. To see the cumulative column for many values of p at once, build a full binomial distribution table.
What Is the Binomial CDF?
The binomial CDF, or cumulative distribution function, gives the probability that a binomial random variable X is at most k: F(k) = P(X ≤ k). It is the running total of the probabilities P(X = 0), P(X = 1), up to P(X = k).
The binomial distribution counts successes in n independent trials that each succeed with the same probability p. The probability of exactly k successes is C(n, k) pk (1 − p)n−k, and the CDF adds those values from 0 up to k. The binomial distribution calculator shows the full distribution with its mean, variance and shape.
Three properties are worth remembering. The CDF never decreases as k grows. It starts at P(X = 0) and reaches exactly 1 at k = n. And it is a step function, because X only takes whole-number values, so F jumps at each integer and stays flat in between.
What Is the Binomial CDF Formula?
The cumulative probability is the sum, for j from 0 to k, of C(n, j) pj (1 − p)n−j.
There is no shorter closed form, so every tool simply adds the terms or uses an equivalent incomplete beta function. For small k the sum is quick. For large n and a k near the top, it is faster to use the complement:
The complement is also how “at least” questions are normally solved by hand. This calculator sums the smaller side directly rather than subtracting from 1, so it stays accurate even when the answer is tiny. The binomial coefficients in each term come from row n of Pascal's triangle, which the binomial expansion calculator lists in full.
How Do You Turn Wording Into the Right Cumulative Probability?
Most mistakes with cumulative binomial questions come from the boundary. Translate the sentence into a symbol first, then pick the matching mode.
| Wording | Symbol | Sum of terms | In terms of F |
|---|---|---|---|
| at most k, k or fewer, no more than k | P(X ≤ k) | 0 to k | F(k) |
| fewer than k, less than k | P(X < k) | 0 to k − 1 | F(k − 1) |
| at least k, k or more, no fewer than k | P(X ≥ k) | k to n | 1 − F(k − 1) |
| more than k, greater than k | P(X > k) | k + 1 to n | 1 − F(k) |
| between a and b inclusive | P(a ≤ X ≤ b) | a to b | F(b) − F(a − 1) |
“At most 4” includes 4. “Fewer than 4” stops at 3. That one-step difference is the most common off-by-one error, and it changes the answer. For 10 fair flips, P(X ≤ 4) = 0.3770 but P(X < 4) = P(X ≤ 3) = 0.1719.
Worked Example: At Most 4 Heads in 10 Flips
Flip a fair coin 10 times. What is the probability of at most 4 heads? Here n = 10, p = 0.5 and k = 4.
The individual probabilities are 0.0010, 0.0098, 0.0439, 0.1172 and 0.2051 for 0 to 4 heads. Their sum is 0.376953125, so P(X ≤ 4) ≈ 0.3770, or 37.70%.
The complement is P(X ≥ 5) = 0.6230. By symmetry, P(X ≥ 7) = P(X ≤ 3) = 0.1719, which is the chance of at least 7 heads. Enter n = 10, p = 0.5, k = 4 and choose “At most” to see the same terms. The exact probability of 5 heads is 0.2461, and it is the largest single term.
Worked Example: Defective Items in a Batch of 20
A process makes 5% defective parts. You inspect 20 independent parts. What is the probability of at most 1 defective? Here n = 20, p = 0.05 and k = 1.
P(X = 0) = 0.9520 = 0.3585 and P(X = 1) = 20 × 0.05 × 0.9519 = 0.3774. Added together, P(X ≤ 1) = 0.7358.
So the batch passes a “one defective allowed” rule about 73.6% of the time, and fails about 26.4% of the time, because P(X ≥ 2) = 0.2642. If the parts come from a small lot and are not replaced, use the hypergeometric calculator instead, since the trials are no longer independent.
Worked Example: Guessing on a Multiple-Choice Test
A 20-question test has four options per question, and a student guesses every answer. Each guess is right with p = 0.25. What is the chance of getting at least 10 right? Here n = 20 and k = 10.
Adding 11 terms from k = 10 to 20 is tedious, so use the complement: P(X ≥ 10) = 1 − P(X ≤ 9) = 1 − 0.9861 = 0.0139. A pure guesser reaches 10 out of 20 only about 1.4% of the time.
The expected score is 20 × 0.25 = 5 correct, with a standard deviation of 1.94. At most 5 correct has probability 0.6172, so more than half the guessers land at 5 or below.
How Do You Find the Probability Between Two Values?
Subtract two cumulative probabilities. For an inclusive range, P(a ≤ X ≤ b) = F(b) − F(a − 1). Subtracting F(a) instead would leave out the value a itself.
Take 12 rolls of a die and count the sixes, so n = 12 and p = 1/6. The probability of between 1 and 3 sixes inclusive is F(3) − F(0) = 0.8749 − 0.1122 = 0.7627. The calculator does the same sum term by term and lists P(X = 1), P(X = 2) and P(X = 3) in the working.
If the question says “between 1 and 3” and means the endpoints are excluded, only 2 counts. Read the wording carefully and write the symbols before you type any numbers.
How Do You Find k From a Probability (Inverse Binomial)?
The inverse binomial finds the smallest k with P(X ≤ k) at least a target probability. It is the discrete counterpart of a percentile, and the tool is useful for setting cut-offs such as “how many defects before we reject the batch?”.
With n = 100 and p = 0.1, the cumulative probabilities near the top are F(14) = 0.9274, F(15) = 0.9601 and F(16) = 0.9794. For a 95% target, the answer is k = 15, because F(14) is still below 0.95 and F(15) is the first value above it. The median is k = 10.
Because X is discrete, the cumulative probability at the answer usually overshoots the target, here 0.9601 against 0.95. The calculator shows both F(k) and F(k − 1) so you can see where the target falls between them. Switch to inverse mode, enter n = 100, p = 0.1 and 95% to try it.
What Is the Difference Between the Binomial CDF and PMF?
The PMF gives the probability of exactly k successes. The CDF gives the probability of k or fewer. On a TI-84 these are binompdf and binomcdf.
| Feature | PMF | CDF |
|---|---|---|
| Question answered | Exactly k successes | k or fewer successes |
| Formula | C(n, k) pk (1 − p)n−k | Sum of the PMF from 0 to k |
| Shape | Bars that rise and fall | Steps that only climb to 1 |
| Calculator button | binompdf | binomcdf |
| Excel last argument | FALSE | TRUE |
For 10 fair flips the PMF peaks at 0.2461 for 5 heads, while the CDF passes 0.5 between 4 and 5 heads and reaches 1 at 10. The chart in the calculator draws both, with the bars for the PMF and a line for the cumulative total. For single-count and range probabilities with a formula-first layout, the binomial probability calculator covers that side.
How Do You Use binomcdf on a TI-84, and in Excel, R and Python?
Every tool has a function for the cumulative probability. They all return P(X ≤ k), and the differences are in the argument order.
| Question | TI-84 | Excel / Sheets | R | Python (SciPy) |
|---|---|---|---|---|
| P(X = k) | binompdf(n, p, k) | =BINOM.DIST(k, n, p, FALSE) | dbinom(k, n, p) | binom.pmf(k, n, p) |
| P(X ≤ k) | binomcdf(n, p, k) | =BINOM.DIST(k, n, p, TRUE) | pbinom(k, n, p) | binom.cdf(k, n, p) |
| P(X ≥ k) | 1 − binomcdf(n, p, k−1) | =1-BINOM.DIST(k-1, n, p, TRUE) | pbinom(k-1, n, p, lower.tail=FALSE) | binom.sf(k-1, n, p) |
| Smallest k with F(k) ≥ q | invBinom(q, n, p) (newer models) | =BINOM.INV(n, p, q) | qbinom(q, n, p) | binom.ppf(q, n, p) |
On a TI-84 the key sequence is 2nd, VARS, then binomcdf( in the list. Enter n, p and k separated by commas. Excel's BINOM.INV is the inverse mode in this calculator, and it needs the target probability in the third argument.
How Accurate Is a Normal Approximation to the Binomial CDF?
When n is large and p is not extreme, the binomial CDF can be approximated by a normal CDF with mean np and standard deviation √(np(1 − p)). Use a continuity correction, because the binomial is discrete and the normal curve is continuous.
For n = 100 and p = 0.5, the exact P(X ≤ 45) is 0.1841. The normal approximation with the correction is Φ((45.5 − 50)/5) = 0.1841, and without the correction it is Φ(−1) = 0.1587, which is clearly off. The normal CDF calculator and the normal curve calculator can run the approximation for you.
The rule of thumb is that np and n(1 − p) should both be at least 10. This page uses the exact sum at every n, so there is no approximation error. For rare events with large n and small p, such as n = 200 and p = 0.02, the Poisson distribution calculator is the usual shortcut.
What Mistakes Do People Make With Binomial CDF Problems?
- Off-by-one at the boundary. “At least 10” is 1 − F(9), not 1 − F(10). A tool that gives P(X ≤ k) will not subtract the extra term for you.
- Using the PMF when the question is cumulative. binompdf gives exactly k, so it cannot answer “at most”.
- Entering a percentage as a whole number. p = 25 is not 25%. Use 0.25 or type 25%.
- Using the binomial when the trials are dependent. Drawing cards without replacement is hypergeometric. Waiting for the first success is geometric.
- Treating the inverse as exact. The smallest k with F(k) ≥ 0.95 rarely gives exactly 0.95. Report both F(k) and F(k − 1).
- Subtracting the wrong CDF values for a range. For inclusive bounds a to b, subtract F(a − 1), not F(a).
Where Is the Binomial CDF Used?
- Quality control. Accept a lot if the number of defects in a sample is at most c. The CDF gives the acceptance probability.
- Hypothesis tests. The exact binomial test builds its p-value from tail sums such as P(X ≥ observed).
- A/B tests and polls. How surprising is 60 successes out of 100 when the true rate is 50%? It is P(X ≥ 60) = 0.0284.
- Reliability. A system works if at least m of n components work, which is a right-tail binomial sum.
- Game design. The chance that a drop rate yields at least one item in n attempts is 1 − F(0).
Frequently Asked Questions
What is a binomial CDF calculator?
It computes the cumulative probability P(X ≤ k) for a binomial distribution by adding the probabilities of 0 through k successes. The same tool handles fewer than, at least, more than and between-two-values questions.
What does binomcdf do?
binomcdf(n, p, k) on a TI-83 or TI-84 returns the probability of k or fewer successes in n trials with success probability p. It equals the sum of binompdf(n, p, j) for j from 0 to k.
How do I find P(X ≥ k) with a binomial CDF?
Use the complement: P(X ≥ k) = 1 − P(X ≤ k − 1). For 10 fair coin flips, P(X ≥ 7) = 1 − P(X ≤ 6) = 1 − 0.8281 = 0.1719.
What is the difference between P(X ≤ k) and P(X < k)?
P(X ≤ k) includes k and P(X < k) stops at k − 1, so P(X < k) = P(X ≤ k − 1). With n = 10 and p = 0.5, P(X ≤ 4) = 0.3770 and P(X < 4) = 0.1719.
How do I find the probability between two values?
Subtract two cumulative values: P(a ≤ X ≤ b) = F(b) − F(a − 1). For n = 12 and p = 1/6, the probability of 1 to 3 sixes is 0.7627.
What is the inverse binomial CDF?
It returns the smallest k for which P(X ≤ k) is at least a chosen probability. For n = 100, p = 0.1 and a target of 95%, the answer is k = 15, because F(14) = 0.9274 and F(15) = 0.9601.
How big can n be in this calculator?
n can be any whole number up to 100,000. The probabilities are summed from the exact binomial terms, so the answer does not depend on a normal or Poisson approximation.
How accurate is this binomial CDF calculator?
It sums the smaller tail directly in double precision, so very small probabilities keep their significant digits. Results agree with SciPy's binom.cdf, binom.sf and binom.ppf across the tested range.
Sources: NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.6.18 Binomial Distribution. SciPy documentation, scipy.stats.binom. R documentation, Binomial. Microsoft Support, BINOM.DIST and BINOM.INV functions.
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