Binomial PDF & CDF
Calculator
Calculate P(X = k), cumulative probabilities, and ranges in one place. Use the distribution and formula steps to interpret the result.
Binomial PDF & CDF Calculator
Enter your parameters below — results update as you type.
Looking for a binompdf or binomcdf calculator? Choose an exact, cumulative, upper-tail, or range probability below. This tool calculates binomial probabilities; it does not expand the binomial theorem.
View full probability table
| k | P(X = k) | P(X ≤ k) | P(X ≥ k) |
|---|
The chart highlights outcomes included in your result. Above 220 outcomes, bars group consecutive values. For exact values, open the full probability table.
Enter parameters and click Calculate to see the full derivation.
Made to explain the calculation.
Designed for practice.
The binomial formula
The calculator displays the binomial coefficient C(n,k) × pk × (1−p)n−k and the steps used to read the result.
Source: NIST/SEMATECH Engineering Statistics HandbookUse it in the browser
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Privacy information should be reviewed before useFor common statistics exercises
The page includes AP Statistics and TI-84 notation. Confirm the notation, rounding, and permitted tools required by your own course or exam.
Check your course or exam requirementsMore than a number
Every result includes an interactive distribution chart, mean, variance, standard deviation, and a full probability table — so you understand the context, not just the answer.
Use the chart and table to compare outcomesMatch the wording to the probability
The numbers may stay the same, but the wording changes the calculation. Choose the result that answers the question you were given.
Exactly k
One exact outcome: P(X = k), such as exactly 3 successes.
At most k
k or fewer: P(X ≤ k).
At least k
k or more: P(X ≥ k).
Between two values
An inclusive range: P(k₁ ≤ X ≤ k₂).
Check and understand your statistics exercise
Exam-ready output format
Results include the probability, formula, and summary statistics. Follow the rounding and notation required by your course. Step-by-step derivation shows C(n,k), the formula substitution, and the final answer — copy it directly into your homework or exam prep notes.
TI-84 mirror
See the exact keystrokes to replicate your result on a TI-84 Plus. No more hunting through menus.
Full probability table
One click reveals P(X=k), P(X≤k) and P(X≥k) for every value of k from 0 to n. Exportable as CSV.
Live distribution chart
The bar chart updates in real time as you change n and p. See how the shape shifts from skewed to symmetric.
Range probability
Calculate P(k₁ ≤ X ≤ k₂) in one step — with both endpoints included.
From parameters to answer
in 4 steps
Enter n — number of trials
This is the total number of independent experiments. Flipping a coin 20 times? n = 20. Rolling a die 6 times? n = 6. Must be a positive integer.
Set p — probability of success
The probability of the outcome you care about on a single trial. Fair coin → p = 0.5. Rolling a specific face on a die → p = 1/6 ≈ 0.167. Use the slider for quick tuning.
Choose k and calculation type
k is the number of successes you're asking about. Then pick: exactly k, at most k (CDF), at least k (survival), or between two values. Each maps directly to a TI-84 function.
Read results and formula
The probability appears instantly with the full step-by-step derivation, a distribution chart highlighting your k value, and all summary statistics (mean, variance, std dev).
What is the Binomial PDF? A complete guide
The binomial probability mass function (PMF) gives P(X = k), the probability of exactly k successes in n independent trials with the same success probability p. Calculators commonly label it binomial PDF or binompdf. The CDF gives P(X ≤ k), the probability of k or fewer successes.
The binomial PMF formula (binompdf)
Where C(n, k) = n! / (k! · (n−k)!) is the binomial coefficient — the number of ways to choose k successes from n trials. The formula multiplies this by the probability of getting exactly k successes and exactly n−k failures in any specific order.
binompdf vs binomcdf: exact vs cumulative probability
For an exact count use binompdf; for a cumulative count use binomcdf: the two functions are closely related but answer different questions.
- Binomial PDF — P(X = k): The probability of getting exactly k successes. Use this when you want a single specific outcome. On a TI-84:
binompdf(n, p, k). - Binomial CDF — P(X ≤ k): The cumulative probability of getting k or fewer successes. It sums the PDF from 0 to k. On a TI-84:
binomcdf(n, p, k). - Survival function — P(X ≥ k): The probability of k or more successes. Calculated as 1 − P(X ≤ k−1). Useful for quality control and hypothesis testing.
How to use binompdf and binomcdf on a TI-84
The TI-84 has a built-in binomial PDF function accessible via the DISTR menu. Here are the exact steps:
- Press 2nd then VARS to open the DISTR menu.
- Scroll down to binompdf( (option A) and press ENTER.
- Enter the parameters in order:
binompdf(n, p, k). Example:binompdf(10, 0.4, 3). - Press ENTER to see the result. For the CDF, use binomcdf( instead (option B).
Our calculator above shows you the TI-84 equivalent for any calculation you run — click the "TI-84 equivalent" button after computing a result.
When does a binomial distribution apply?
A random variable X follows a binomial distribution B(n, p) when all four BINS conditions are met:
- Binary outcomes — each trial results in success or failure.
- Independent trials — the outcome of one trial doesn't affect others.
- Number of trials is fixed — n is determined before the experiment.
- Same probability — p is identical across all trials.
Worked binomial probability examples
A factory produces parts where 5% are defective (p = 0.05). In a batch of 20 (n = 20), what's the probability exactly 2 are defective? P(X = 2) = C(20,2) × 0.05² × 0.95¹⁸ ≈ 0.1887.
Calculate this exampleA student guesses randomly on a 10-question multiple choice test (4 options each, p = 0.25). What's the probability of passing with at least 6 correct? P(X ≥ 6) ≈ 0.0197.
Calculate this exampleA drug is effective for 70% of patients (p = 0.7). In a trial with 15 patients (n = 15), what's the probability exactly 10 respond? P(X = 10) = C(15,10) × 0.7¹⁰ × 0.3⁵ ≈ 0.2061.
Calculate this exampleMean, variance, and standard deviation of the binomial distribution
| Statistic | Formula | For n=10, p=0.4 |
|---|---|---|
| Mean (μ) | n · p | 4.00 |
| Variance (σ²) | n · p · (1 − p) | 2.40 |
| Std Dev (σ) | √(n · p · (1 − p)) | 1.549 |
As n increases, the binomial distribution approaches the normal distribution (by the Central Limit Theorem), which is why a normal approximation is valid when both np ≥ 10 and n(1−p) ≥ 10.
Can you do binomial PDF on a TI-30XS?
The TI-30XS MultiView does not have a built-in binompdf() function — it's a scientific calculator, not a graphing calculator. However, you can compute the binomial probability manually using the combination function: press PRB and select nCr, then multiply by p^k and (1−p)^(n−k). You can use this browser calculator to check your manual binomial calculation.
Check that the situation is binomial
A binomial model applies when the number of trials is fixed, each trial has two possible outcomes, trials are independent, and the probability of success stays constant.
The calculator can work from the values you enter, but it cannot decide whether those conditions hold in a real situation. Take care with sampling without replacement or with a probability that changes from trial to trial.
Frequently asked questions
What does binomial PDF calculate exactly?
The binomial PDF calculates the probability of getting exactly k successes in n independent trials, where each trial has probability p of success. It uses the formula P(X = k) = C(n,k) × p^k × (1−p)^(n−k). "PDF" stands for probability density function, though for discrete distributions like the binomial, it's technically a probability mass function (PMF).
Where is binomial PDF on a calculator? (TI-84 step-by-step)
On a TI-84 or TI-84 Plus CE: press 2nd + VARS (DISTR), scroll to binompdf( and enter binompdf(n, p, k). For the cumulative version, use binomcdf(. Our calculator mirrors these functions and shows you the exact TI-84 commands for every result you compute.
What is the difference between binomial PDF and CDF?
What are the conditions for using the binomial distribution?
Remember BINS: Binary outcomes (success/failure only), Independent trials (one outcome doesn't influence another), Number of trials is fixed before the experiment, and Same probability of success p across all trials. If any condition is violated, consider other distributions (hypergeometric if sampling without replacement; negative binomial if counting trials until k successes).
How do I calculate P(X ≥ k) — "at least k successes"?
Can this calculator handle large n (e.g., n = 500 or n = 1000)?
Yes. This calculator uses logarithm-based computation (log-gamma functions) to avoid overflow for large n values, up to n = 1000. For very large n where np ≥ 10 and n(1−p) ≥ 10, the normal approximation is also a practical option: X ≈ N(np, np(1−p)).
What is the mean and standard deviation of a binomial distribution?
For a binomial distribution B(n, p): the mean (expected value) is μ = n × p, the variance is σ² = n × p × (1−p), and the standard deviation is σ = √(n × p × (1−p)). Our calculator displays all three for any parameters you enter, along with the mode (most likely number of successes).
Is this calculator free? Do I need to sign up?
You can use the calculator without creating an account. Check the site privacy notice for the current data and cookie configuration; this page does not make an offline-use claim.
How is binomial PDF different from the normal distribution?
The binomial distribution is discrete (integer values of k only) and applies when you have a fixed number of trials with binary outcomes. The normal distribution is continuous and applies to measurements. However, when n is large and p is not too extreme, the binomial is well-approximated by a normal distribution with μ = np and σ = √(np(1−p)) — this is the normal approximation to the binomial.
Can I use this for negative binomial or Poisson distribution?
This calculator is specifically for the standard binomial distribution. The negative binomial distribution counts the number of trials needed to get k successes (a different setup). The Poisson distribution applies when events happen at a constant rate over time or space. These require separate calculators with different formulas.