Binomial Probability Calculator
Calculate exact, less than, greater than, and interval probabilities using execution frequency and success probability. Save an exponential distribution graph and a total probability table.
Input
Calculate directly using the sample value, or try changing the value.
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Result
Check the result using the example value.
Calculation·solution process
Verification of calculation data
The display value is a rounded approximation of 10 digits. CSV or TXT display values are saved; input values are not included in addresses or visit statistics.
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Calculation criteria and usage method
Determining the exact, below, and above success probabilities for independent execution.
Input example
When tossing a fair coin 10 times, the probability that the head appears exactly 5 times is 252/1,024=24.609375%. Less than 5 is 62.3046875%, and more than 5 is also 62.3046875%.
If n=10, p=0.5, and k=5, then the number of combinations C(10,5)=252, and the probability for each case is 1/1,024. Exactly 5 times is 252/1,024=24.609375%. Below 5 and above 5 is 62.3046875% each, and since both include 5, adding the two values exceeds 100%.
How to use
- Use the example value or enter numbers directly. The table format has one row per line.
- Click Verify result. Incorrect rows are marked and excluded.
- Check the result and calculation process, then copy or save the required result as a file.
Limitations and interpretation
n=0~10,000, p=0~1, k and the interval is 0~n integers. Two outcomes: success or failure. Independent trials with the same success probability each time. No normal approximation is used; extremely small probabilities are represented as log values to indicate limits.
Frequently Asked Questions
How is the success probability of 50% entered?
Enter 0.5 for p. n and k are integers with 0≤k≤n. If the end of the range is left empty, calculate up to n attempts; all specified ends are included.
When can binomial distribution be used?
Used when there are two results: success and failure, a fixed execution count, independent executions, and the same success probability each time. This calculation cannot be directly applied to successes or failures that affect each other or change over time.
Calculation criteria and reference materials
- NIST — binomial distribution
Probability, mass, cumulative probability, mean and standard deviation
Verification of calculation criteria: · Calculation·verification principles · Report errors