Binomial PDF
Calculator
Instantly compute P(X = k), cumulative probabilities, visualize the full distribution, and understand every step — the most complete binomial tool available online.
Built for accuracy.
Designed for students.
Exact formula, every time
We use the binomial coefficient C(n,k) × pk × (1−p)n−k with full floating-point precision. Every calculation is shown step-by-step so you can verify it yourself.
Source: NIST Digital Library of Mathematical Functions §26.4100% browser-side
All calculations happen locally in your browser. Nothing is sent to any server. No cookies, no tracking, no account required. Open the source with DevTools and verify it yourself.
Privacy by design — zero data collectedTrusted by students
Aligned with AP Statistics, introductory college probability, and actuarial exam prep. The output format mirrors Texas Instruments TI-84 binompdf() and binomcdf() functions.
Compatible with AP Stats, Stat 101, actuarial SOA/CASMore 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.
Visual learning backed by cognitive scienceBinomial PDF & CDF Calculator
Enter your parameters below — results update as you type.
Enter parameters and click Calculate to see the full derivation.
View full probability table
| k | P(X = k) | P(X ≤ k) | P(X ≥ k) |
|---|
Everything you need to
ace your stats exam
Exam-ready output format
Results are displayed exactly as expected in AP Statistics and college-level courses. 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 — a feature most online calculators don't offer.
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 density function (technically called the probability mass function for discrete distributions, though "PDF" is commonly used interchangeably) gives you the exact probability of obtaining exactly k successes in n independent trials, where each trial has the same probability p of success.
The binomial PDF formula
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.
Binomial PDF vs CDF — what's the difference?
This is one of the most searched questions in introductory statistics, and for good reason: 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 find binomial PDF 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.
Real-world examples of binomial PDF calculations
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.
Try this →A 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.
Try this →A 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.
Try this →Mean, 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). Our web calculator is a perfect TI-30XS substitute for binomial distribution problems.
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?
The binomial PDF gives you P(X = k) — the probability of exactly k successes. The CDF (cumulative distribution function) gives you P(X ≤ k) — the probability of k or fewer successes, which is the sum of PDF values from 0 to k. Use PDF when you need an exact count; use CDF when you need "at most" or "at least" probabilities.
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"?
P(X ≥ k) = 1 − P(X ≤ k−1). In other words, subtract the CDF at k−1 from 1. On a TI-84: 1 − binomcdf(n, p, k−1). Our calculator handles this automatically when you select the "P(X ≥ k)" option — just enter your k and hit calculate.
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?
Yes, completely free. No account, no signup, no email required. All calculations run directly in your browser — no data is sent to any server. You can even bookmark this page and use it offline after the first load.
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. We plan to add both as separate tools — check back soon.