In a certain state lottery, people select eight numbers in advance of a random drawing of six numbers. If someone's selections include the six drawn, they receive a large prize, but this prize is shared with other winners. Victor has discovered that some numbers are "unpopular" in that they are rarely chosen by lottery players. He has computed that by selecting these numbers he has one chance in a million of winning \(\$ 10\) million for a \(\$ 1\) lottery ticket. He has odds of 10 to 1 in his favor. Victor's current wealth is \(\$ 100,000\), and he wants to maximize the expected logarithm of wealth.

(a) Should Victor buy a lottery ticket?

(b) Victor knows that he can buy a fraction of a ticket by forming a pool with friends. What fraction of a ticket would be optimal?