2014
Problem - 4589
Moor has $\$1000$, and he is playing a gambling game. He gets to pick a number k between $0$ and $1$ (inclusive). A fair coin is then flipped. If the coin comes up heads, Moor is given $5000k$ additional dollars. Otherwise, Moor loses $1000k$ dollars. Moor’s happiness is equal to the log
of the amount of money that he has after this gambling game. Find the value of k that Moor
should select to maximize his expected happiness.
Assuming that Moor chooses a value of $k$, then his expected value of happiness is
$$f(k) = \frac{1}{2}\log(1000+5000k)+\frac{1}{2}\log(1000-1000k)=\frac{1}{2}\left(6 + \log(5k+1)(1-k)\right)$$
This function reaches maximum when $(5k+1)(1-k)$ does so. Hence, the answer is $\boxed{\frac{2}{5}}$.