Enter your values
Use a probability between 0 and 1. The complement probability is calculated automatically.
Example data table
These sample values use base 2, so entropy is shown in bits.
| Probability p | Complement q | Binary Entropy H(p) | Interpretation |
|---|---|---|---|
| 0.05 | 0.95 | 0.286397 | Low uncertainty |
| 0.10 | 0.90 | 0.468996 | Moderate uncertainty |
| 0.25 | 0.75 | 0.811278 | High uncertainty |
| 0.50 | 0.50 | 1.000000 | Maximum uncertainty zone |
| 0.75 | 0.25 | 0.811278 | High uncertainty |
| 0.90 | 0.10 | 0.468996 | Moderate uncertainty |
Formula used
Binary entropy measures the average uncertainty of a two-outcome event. For an event with probability p and complement probability q = 1 - p, the entropy is:
- b is the log base. Use 2 for bits, e for nats, or 10 for hartleys.
- Maximum entropy for a binary source equals logb(2).
- Normalized entropy = H / Hmax.
- Redundancy = 1 - normalized entropy.
- Expected total information = sample size × H.
- Self-information of an outcome = -logb(probability).
When p is 0 or 1, entropy becomes zero because the result is certain. When p is 0.50, entropy is highest because both outcomes are equally likely.
How to use this calculator
- Enter a label for the primary event and its complement.
- Type the event probability p as a decimal between 0 and 1.
- Set a sample size if you want expected counts and total information.
- Choose a log base, or enter a custom base greater than zero and not equal to 1.
- Select the number of decimal places for output formatting.
- Press Calculate Entropy to view the results block above the form.
- Review the metric cards, interpretation label, and entropy curve.
- Use the CSV or PDF buttons to export the result summary or example data.
Frequently asked questions
1) What does binary entropy measure?
Binary entropy measures uncertainty in a two-outcome event. It tells you how unpredictable the result is on average, based on the event probability and its complement.
2) When is entropy highest?
Entropy is highest when p = 0.50. At that point, both outcomes are equally likely, so uncertainty is at its maximum for a binary source.
3) Why does entropy become zero at p = 0 or p = 1?
Entropy becomes zero because the outcome is certain. If one event always happens, there is no uncertainty left to measure.
4) What is the difference between bits, nats, and hartleys?
They are different entropy units based on the log base. Base 2 gives bits, base e gives nats, and base 10 gives hartleys.
5) What does normalized entropy mean?
Normalized entropy compares the calculated entropy with the maximum possible entropy for a binary source. It helps you judge uncertainty on a 0 to 1 scale.
6) What does redundancy show?
Redundancy shows how much predictability remains. A higher redundancy value means lower uncertainty and more structure in the binary process.
7) Why include sample size?
Sample size lets you estimate expected counts and total information over many trials. It turns a per-event entropy value into a practical planning metric.
8) Can I use this for machine learning or coding problems?
Yes. Binary entropy is useful in information theory, compression, decision trees, classification analysis, communication systems, and probability-based optimization tasks.