I was reading a tutorial introduction to Information Theory and it presented a formula for determining 'average surprise' of 100 coin flips with 50 heads and 50 tails, as follows:
$${[50 * log(1/0.5)] + [50 * log(1/0.5)] \over 100}$$
The text goes on to say that the above expression reduces to:
$$[0.5 * log(1/0.5)] + [0.5 * log(1/0.5)]$$
My question is, why did the log(1/0.5) not reduce to a smaller number in the same way that 50 reduces to 0.5 ? I intuitively expect that it should reduce to:
$$(0.5 * {log(1/0.5) \over 100}) + (0.5 * {log(1/0.5) \over 100})$$
More generally, how are logs in the numerator simplified? Thank you.
First, you can do $$\frac {50a+50b}{100}=\frac {50(a+b)}{100}=0.5(a+b)$$ Then $1/0.5=2$ so I don't know why the book did not substitute $$\log \frac 1{0.5}=\log 2$$ to get $$\log 2$$ as the final answer.