You have an opportunity to place a bet on the outcome of an upcoming race involving a certain female

horse named Bayes: if you bet x dollars and Bayes wins, you will have w0 + x, while if she loses you

will have w0 x, where w0 is your initial wealth.

(a) Suppose that you believe the horse will win with probability p and that your utility for wealth w

is ln(w). Find your optimal bet as a function of p and w0.

(b) You know little about horse racing, only that racehorses are either winners or average, that winners

win 90% of their races, and that average horses win only 10% of their races. After all the buzz

you've been hearing, you are 90% sure that Bayes is a winner. What fraction of your wealth do

you plan to bet?

(c) As you approach the betting window at the track, you happen to run into your uncle with a

gambling problem. He knows rather a lot about horse racing: he correctly identifies a horse's true

quality 95% of the time. You relay your excitement about Bayes. "Don't believe the hype" he

states. "That Bayes mare is only an average horse." What do you bet now (assume that the rules

of the track permit you to receive money only if the horse wins)?