Binary Entropy Calculator: Formula and Meaning

A binary entropy calculator finds the Shannon entropy of a two-outcome variable from the probability of either outcome. Enter the probability of one outcome as a number from 0 to 1; the calculator returns entropy in bits, along with the probability of the other outcome.

Formula

The binary entropy formula is H(p) = −p log₂(p) − (1 − p) log₂(1 − p). Here, p is the probability of outcome 1, and 1 − p is the probability of outcome 0. Probability has no unit; entropy is measured in bits when the logarithm has base 2.

The formula describes the average information, or uncertainty, associated with one observation of a binary variable. A probability of zero or one means the outcome is certain, so entropy is zero. At p = 0 or p = 1, the expression includes a logarithm of zero; by convention, the corresponding term is defined by its limit as zero. A calculator should handle these endpoints directly rather than treating them as errors.

Worked Example

With the default input, p = 0.5. The complementary probability is q = 1 − 0.5 = 0.5. Substituting into the formula:

H = −0.5 log₂(0.5) − 0.5 log₂(0.5)

Since log₂(0.5) = −1, H = 0.5 + 0.5 = 1.0000 bit. The output is rounded to four decimal places, as specified in the calculator block.

What Binary Entropy Measures

Entropy is a measure of uncertainty before a result is observed, not a measure of how surprising a particular result feels after it happens. For a binary variable, the probabilities of the two outcomes must add to 1. If one outcome has probability 0.8, the other has probability 0.2, and the entropy is about 0.7219 bits. The outcome with probability 0.8 is more likely, so the result is less uncertain than an even split.

The function is symmetric: H(p) = H(1 − p). Thus, probabilities 0.2 and 0.8 give the same entropy. Its maximum is 1 bit at p = 0.5, where the two outcomes are equally likely. Its minimum is 0 bits at either endpoint, where the outcome is certain.

The word “bit” here is an information unit. An entropy of 1 bit does not mean that every outcome can always be represented using exactly one physical bit in every coding system. It gives the theoretical average uncertainty per independent observation; practical encoding also depends on the data and coding method.

How to Use the Calculator

1. Enter the probability of one of the two outcomes in the range 0 to 1. For example, use 0.25 for a 25% chance.
2. Read the entropy output in bits. The second output shows the complementary probability, which is 1 minus the value entered.
3. Check that the probability represents the outcome and population or process you actually want to describe.

If your input is a percentage, convert it to a probability first: divide by 100. For example, 35% becomes 0.35. Do not enter 35 as though it were a probability. The calculator uses a dimensionless probability, so no physical unit conversion is needed.

Interpreting Examples and Applications

Here are several reference points, rounded to four decimal places:

• p = 0 or 1: H = 0 bits; the outcome is certain.
• p = 0.1 or 0.9: H ≈ 0.4690 bits.
• p = 0.2 or 0.8: H ≈ 0.7219 bits.
• p = 0.5: H = 1 bit; uncertainty is greatest for a binary variable.

Binary entropy is used in information theory, data compression, and the analysis of binary sources. It also appears in machine learning as a measure of uncertainty in a two-class probability distribution. In these settings, entropy can help compare distributions, but it does not by itself say whether a prediction is correct, whether a data source is biased, or how well a model performs.

For observations gathered from data, an estimated probability such as the fraction of outcomes coded as 1 can be substituted into the formula. The resulting entropy describes that estimated distribution. It is not a guarantee that the estimate precisely reflects the underlying process, particularly when the sample is small or unrepresentative.

Common Mistakes and Limits

• Using percentages as whole numbers: enter 50% as 0.5, not 50.
• Using the wrong logarithm base: base 2 gives bits. Natural logarithms give the same entropy in nats; convert by dividing by ln(2) to express nats in bits.
• Forgetting the second outcome: for a binary variable, its probability is 1 − p, not an independent value chosen without checking the total.
• Rounding too early: retain the original probability through the calculation and round only the displayed result. Near 0 and 1, small input changes can affect the result, so avoid interpreting extra displayed digits as extra certainty.

Binary entropy applies to a variable with two possible outcomes and a specified probability distribution. If there are three or more outcomes, use the general Shannon entropy formula, H = −Σ pᵢ log₂(pᵢ), with all outcome probabilities included. Entropy also does not capture relationships between successive observations: if outcomes are correlated, entropy per observation alone may not describe the full information rate of the sequence.

Frequently Asked Questions

What is the maximum binary entropy?
It is 1 bit, reached when both outcomes have probability 0.5. That is the most uncertain distribution for a variable with exactly two outcomes.

Why is entropy zero when p is 0 or 1?
At either endpoint, one outcome is certain and the other has probability zero. There is no uncertainty about which outcome will occur, so the entropy is zero.

Can I enter a percentage into a binary entropy calculator?
Convert the percentage to a decimal probability first. For example, 70% should be entered as 0.70.

Is binary entropy the same as probability?
No. Probability describes how likely an outcome is; entropy summarizes the uncertainty of the whole two-outcome distribution. Different probabilities can produce the same entropy because H(p) equals H(1 − p).

Why does the logarithm base change the units?
The base determines the information unit. Base 2 measures entropy in bits, while the natural logarithm measures it in nats. The underlying uncertainty is the same; only the numerical scale changes.

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in p | Probability of outcome 1 | - | 0.5 | 0
out H | Binary entropy | bits | -p * log2(p) - (1 - p) * log2(1 - p) | 4
out q | Probability of outcome 0 | - | 1 - p | 2