1. What is Entropy?

Entropy is a measure of uncertainty or randomness in a system. In information theory, it quantifies the expected value of the information contained in a message. In physics, it's a measure of the number of specific ways a thermodynamic system may be arranged.

2. How Does the Calculator Work?

The calculator uses the entropy formula:

Where:

• \( S \) — Entropy (in nats)
• \( p_i \) — Probability of state i
• \( \ln \) — Natural logarithm

Explanation: The equation sums the product of each probability with its natural logarithm, then takes the negative of that sum.

3. Importance of Entropy Calculation

Details: Entropy is fundamental in information theory, thermodynamics, statistical mechanics, and machine learning. It helps quantify disorder, uncertainty, and information content.

4. Using the Calculator

Tips: Enter probabilities between 0 and 1 separated by commas. The probabilities will be normalized to sum to 1 if they don't already. All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What units does entropy use?
A: When using natural logarithm (ln), the unit is "nats". For base-2 logarithm, the unit would be "bits".

Q2: What is the maximum possible entropy?
A: For N states, maximum entropy is ln(N) when all states are equally probable.

Q3: What does zero entropy mean?
A: Zero entropy means the system is perfectly predictable (one outcome has probability 1, others 0).

Q4: How is entropy used in machine learning?
A: It's used in decision trees (information gain), clustering evaluation, and probabilistic models.

Q5: What's the difference between entropy and cross-entropy?
A: Cross-entropy measures the difference between two probability distributions, while entropy measures uncertainty in a single distribution.