What Is Binomial Distribution Table?
A binomial table lists P(X = k) for every possible number of successes k, along with the cumulative probability P(X ≤ k). Printed tables only cover a few values of n and p; this calculator builds the table for whatever n and p you enter.
The summary includes the mean np, the median, the most likely value, the skewness, the central 95 % range of outcomes and a check on whether a normal approximation is reasonable. A probability chart and a cumulative chart show how the distribution changes with p.
Below you will find the formulas, a variables table, a worked example and answers to common questions such as how to read a binomial table and when to use the normal approximation.
How to Use the Binomial Distribution Table Calculator
- Enter n (number of trials) and p (probability of success on each trial).
- Read the summary figures and check the chart.
- Scroll the table to the value of k you need. Use the right-tail column for “at least” questions.
- To work out a single probability with full steps, use the binomial probability with steps calculator.
Binomial Distribution Table Formula
PMF
CDF
Skewness
Binomial Distribution Table Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| n | Number of trials | |
| p | Success probability | Same on every trial |
| P(X = k) | Probability mass function | C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ |
| P(X ≤ k) | Cumulative probability | Sum of the PMF up to k |
Binomial Distribution Table Worked Example
Using the example values (press “Load example” above to try them):
- Each row uses P(X = k) = C(12, k) × 0.35^k × 0.65^(12 − k).
- The cumulative column adds the PMF values from 0 up to k; the right-tail column is 1 minus the previous cumulative value.
- Mean = 12 × 0.35 = 4.2; σ = √(12 × 0.35 × 0.65) = 1.65227.
Result: Mean μ = np = 4.2; Standard deviation σ = 1.65227.
How to read a binomial table
Find the row for your k. The “P(X = k)” column answers “exactly k”. The “P(X ≤ k)” column answers “k or fewer”, which is what a printed cumulative table lists. For “at least k” use the right-tail column, or equivalently 1 − P(X ≤ k − 1). Probabilities for “between a and b inclusive” are P(X ≤ b) − P(X ≤ a − 1).
| Question | Column or formula |
|---|---|
| Exactly k | P(X = k) |
| k or fewer | P(X ≤ k) |
| k or more | P(X ≥ k) |
| Between a and b (inclusive) | P(X ≤ b) − P(X ≤ a − 1) |
| Fewer than k | P(X ≤ k − 1) |
The shape of the binomial
When p = 0.5 the distribution is symmetric. For p below 0.5 it leans to the left with a tail on the right (positive skew); for p above 0.5 the opposite. Larger n makes it more bell shaped, which is why the normal approximation works for large n. The highlighted central 95% shows the range in which the count will fall almost every time.
Where binomial tables are used
- Quality control: how many defects can I expect in a batch?
- Exams: probability of passing a multiple-choice test by guessing.
- Surveys and polls: how many respondents will say yes?
- Medicine: how many patients respond to a treatment?
Where Binomial Distribution Table Is Used
Textbook table look-ups
Reading P(X = k) and P(X ≤ k) for any n and p.
Seeing the shape
Watching how skewness changes as p moves away from 0.5.
Choosing an approximation
Checking whether a normal approximation is acceptable.
Ranges of likely outcomes
Finding the central 95 % range of successes.
Common Binomial Distribution Table Mistakes to Avoid
- Use the right-tail column for upper-tail questions to avoid subtracting by hand.
- For n above 1,000 use the single-value calculator.
- Check np and n(1 − p) before using the normal approximation.
Binomial Distribution Table Terms and Definitions
- PMF
- Probability of exactly k successes.
- Tail probability
- Probability at or beyond a given value.
Binomial Distribution Table FAQs
How do I make a binomial distribution table?
Compute C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ for each k from 0 to n, then add them cumulatively. This calculator does it automatically.
What is the difference between the PMF and CDF columns?
PMF is the probability of exactly k; CDF is the probability of k or fewer.
How do I find the probability of at least k?
Use the P(X ≥ k) column.
How large can n be?
Up to 1,000 for a table. For a single probability with larger n use the binomial probability calculator.
Do the probabilities add to 1?
Yes, the PMF column sums to 1 over all values of k.
What does the central 95% mean?
The smallest range of counts that contains at least 95% of the probability, from the 2.5th to the 97.5th percentile.