Calculator
Example Data Table
Example with n = 5 and p = 0.40.
| k | P(X = k) | P(X ≤ k) |
|---|---|---|
| 0 | 0.07776 | 0.07776 |
| 1 | 0.25920 | 0.33696 |
| 2 | 0.34560 | 0.68256 |
| 3 | 0.23040 | 0.91296 |
| 4 | 0.07680 | 0.98976 |
| 5 | 0.01024 | 1.00000 |
Formula Used
P(X = k) = C(n, k) × pk × (1 - p)n-k
C(n, k) = n! / (k! × (n-k)!)
P(X ≤ k) = Σ P(X = i), for i from 0 to k
P(a ≤ X ≤ b) = Σ P(X = i), for i from a to b
Mean = np
Variance = np(1-p)
Standard deviation = √[np(1-p)]
How to Use This Calculator
- Enter the total number of trials as n.
- Enter the success probability p as a decimal.
- Choose the probability type you want to evaluate.
- Enter k, or the interval limits a and b.
- Set your preferred decimal precision.
- Click Calculate to show results above the form.
- Review the graph, distribution table, and summary measures.
- Use the CSV or PDF buttons to export your output.
About Binomial Probability
What this calculator does
A binomial model counts successes across repeated trials. Each trial has only two outcomes. Success can happen, or it can fail. The probability of success stays fixed. Trials are also independent. This calculator helps you analyze that pattern quickly.
Why n and p matter
The value n tells how many trials occur. The value p tells the chance of success on each trial. These two inputs define the full binomial distribution. Once you know them, you can compute exact probabilities, cumulative probabilities, and ranges of outcomes.
Exact and cumulative answers
Exact probability answers a focused question. It gives the chance of getting one specific number of successes. Cumulative probability answers a broader question. It can show the chance of getting at most a value, less than a value, at least a value, or more than a value. Interval probability helps when you care about a band of outcomes.
Useful summary measures
The mean gives the average number of successes you should expect over many repetitions. The variance shows how spread out the results are. The standard deviation gives that spread in the same unit as the success count. The mode highlights the most likely outcome.
Why the table and chart help
A single probability is helpful, but a full table gives more context. You can see how probability shifts from one success count to another. The chart makes the shape easy to read. Some distributions are centered. Others lean left or right, depending on p.
Common uses
Binomial probability appears in quality control, survey sampling, sales targets, exam guessing, medical testing, and reliability studies. It is useful whenever you repeat the same yes or no process many times. This page combines the formula, the table, the graph, and exports in one place.
FAQs
1. What does n mean in a binomial problem?
n is the number of independent trials. If you flip a coin 12 times, then n equals 12.
2. What does p mean here?
p is the probability of success on one trial. Enter it as a decimal, so 70% becomes 0.70.
3. When should I use exact probability?
Use exact probability when you need one precise outcome, such as exactly 4 successes in 10 trials.
4. What is the difference between at most and less than?
At most includes the target value. Less than does not. So P(X ≤ 3) includes 3, while P(X < 3) stops at 2.
5. Can p be 0 or 1?
Yes. If p is 0, all probability sits at zero successes. If p is 1, all probability sits at n successes.
6. Why does the table total equal 1?
The exact probabilities across all possible outcomes must add to 1. That covers every possible success count.
7. What does the mean tell me?
The mean, equal to np, gives the long-run average number of successes if the same experiment repeats many times.
8. When is a binomial model appropriate?
Use it when trials are independent, only two outcomes exist, the number of trials is fixed, and the success probability stays constant.