Certain outcome: p = 1 — Shannon entropy
coreno uncertainty, no information
Answer
0.0000 bits
Why this example is worth doing
A distribution with a single certain outcome has zero entropy: nothing is learned by observing it. The page uses this to handle the boundary case honestly — the formula contains p log p with p equal to zero for the other symbols, which is undefined as written and defined as zero by the limit. Naive implementations return NaN here, and the page says which convention it uses and why.
Try your own input in the Shannon entropy. Bits per symbol for any distribution, with the surprisal of each symbol shown.
How the answer is reached
Shannon entropy of 1 symbol
1 symbol, 1 total occurrences. Divide by the total to get probabilities, then take −Σ p·log₂ p.
| Symbol | p | Surprisal −log₂ p (bits) | Contribution p·(−log₂ p) |
|---|---|---|---|
| H | 1.000000 | 0.000000 | 0.000000 |
H0.000000 bits/symbol— the probability-weighted mean of the surprisal column
Total information0.000000 bits— 1 symbols × H
Maximum possible H0.000000 bits/symbol— log₂(1), reached only by the uniform distribution
H / log₂(m)1.000000— how close to uniform this source is
A symbol of probability 0 contributes exactly 0: the convention 0·log₂0 = 0 is a definition (the limit as p → 0), not an approximation. Evaluating it instead returns NaN, which is the usual bug on this page.
Source: C. E. Shannon, Bell System Technical Journal 27:379–423 (1948), Theorem 2