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:

and:

Hence the result.