Uncertainty/Examples/Example 1

Example of Uncertainty

Let $R_1$ and $R_2$ be horseraces.

Let $R_1$ have $7$ runners:

$3$ of which each have probability $\dfrac 1 6$ of winning

$4$ of which each have probability $\dfrac 1 8$ of winning.

Let $R_2$ have $8$ runners:

$2$ of which each have probability $\dfrac 1 4$ of winning

$6$ of which each have probability $\dfrac 1 {12}$ of winning.

Then $R_1$ and $R_2$ have equal uncertainty.

Proof

Let $H_1$ and $H_2$ denote the uncertainty of $R_1$ and $R_2$ respectively.

Recall the definition of uncertainty:

$\map H X = \ds -\sum_k p_k \lg p_k$

Thus:

\(\ds H_1\)

\(=\)

\(\ds -\paren {\sum_{k \mathop = 1}^3 \dfrac 1 6 \lg \dfrac 1 6 + \sum_{k \mathop = 1}^4 \dfrac 1 8 \lg \dfrac 1 8}\)

\(\ds \)

\(=\)

\(\ds -\dfrac 1 6 \paren {3 \lg \dfrac 1 6} - \dfrac 1 8 \paren {4 \lg \dfrac 1 8}\)

simplifying

\(\ds \)

\(=\)

\(\ds -\dfrac 1 2 \lg \dfrac 1 6 - \dfrac 1 2 \lg \dfrac 1 8\)

simplifying

\(\ds \)

\(=\)

\(\ds \dfrac 1 2 \paren {\lg 6 + \lg 8}\)

Logarithm of Reciprocal

\(\ds \)

\(=\)

\(\ds \dfrac 1 2 \lg 48\)

Sum of Logarithms

and:

\(\ds H_2\)

\(=\)

\(\ds -\paren {\sum_{k \mathop = 1}^2 \dfrac 1 4 \lg \dfrac 1 4 + \sum_{k \mathop = 1}^6 \dfrac 1 {12} \lg \dfrac 1 {12} }\)

\(\ds \)

\(=\)

\(\ds -\dfrac 1 4 \paren {2 \lg \dfrac 1 4} - \dfrac 1 {12} \paren {6 \lg \dfrac 1 {12} }\)

simplifying

\(\ds \)

\(=\)

\(\ds -\dfrac 1 2 \lg \dfrac 1 4 - \dfrac 1 2 \lg \dfrac 1 {12}\)

simplifying

\(\ds \)

\(=\)

\(\ds \dfrac 1 2 \paren {\lg 4 + \lg 12}\)

Logarithm of Reciprocal

\(\ds \)

\(=\)

\(\ds \dfrac 1 2 \lg 48\)

Sum of Logarithms

Hence the result.

$\blacksquare$

Sources

• 1988: Dominic Welsh: Codes and Cryptography ... (previous) ... (next): $\S 1$: Entropy = uncertainty = information: $1.1$ Uncertainty: Exercises $1.1$: $1.$