A game of chance based on a spinning wheel is available that pays \(n\) times money bet in the case of a win and nothing in the case of a loss. A gambler has developed a device by which he may exercise some control over the wheel in such a way that his chance of winning is \(\alpha / n\) instead of \(1 / n\), where \(\alpha>1\).

(a) Starting with \(\$ 1.00\), how much should the gambler wager to be log optimal?

(b) What is the expected value of the log of wealth after a single play?

(c) Suppose \(n\) is very large and the gambler uses the strategy \(n\) times. What is the expected value of the \(\log\) of final wealth as a function of \(n\) as \(n\) goes to infinity?

(d) Suppose \(n=1\) million and you have \(\$ 1\) million. How much should you bet on the first spin, and how much do you to expect earn after a million spins?