# Uncertainty/Examples/Example 1

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## 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:

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

Thus:

\(\displaystyle H_1\) | \(=\) | \(\displaystyle -\paren {\sum_{k \mathop = 1}^3 \dfrac 1 6 \lg \dfrac 1 6 + \sum_{k \mathop = 1}^4 \dfrac 1 8 \lg \dfrac 1 8}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle -\dfrac 1 6 \paren {3 \lg \dfrac 1 6} - \dfrac 1 8 \paren {4 \lg \dfrac 1 8}\) | simplifying | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle -\dfrac 1 2 \lg \dfrac 1 6 - \dfrac 1 2 \lg \dfrac 1 8\) | simplifying | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac 1 2 \paren {\lg 6 + \lg 8}\) | Logarithm of Reciprocal | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac 1 2 \lg 48\) | Sum of Logarithms |

and:

\(\displaystyle H_2\) | \(=\) | \(\displaystyle -\paren {\sum_{k \mathop = 1}^2 \dfrac 1 4 \lg \dfrac 1 4 + \sum_{k \mathop = 1}^6 \dfrac 1 {12} \lg \dfrac 1 {12} }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle -\dfrac 1 4 \paren {2 \lg \dfrac 1 4} - \dfrac 1 {12} \paren {6 \lg \dfrac 1 {12} }\) | simplifying | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle -\dfrac 1 2 \lg \dfrac 1 4 - \dfrac 1 2 \lg \dfrac 1 {12}\) | simplifying | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac 1 2 \paren {\lg 4 + \lg 12}\) | Logarithm of Reciprocal | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac 1 2 \lg 48\) | 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 $1.1$: $1.$