In a roulette wheel, any number from 0 to 2007 can come up with equal probability. The roulette wheel is spun repeatedly. Let $P_{k}$ denote the probability that at some point the sum of the numbers that have come up in all the spins equals $k$. Which number is greater: $P_{2007}$ or $P_{2008} ?$
Since the sum is zero at the beginning of the game, $P_{0}=1$.

Reasoning in the same way as in problem 65273 (the only difference is that instead of the number 6, the number 2008 is used), we have:

$P_{2007}-P_{2008}=1 / 2008\left(P_{2007}+P_{2006}+\ldots+P_{0}\right)-1 / 2008\left(P_{2008}+P_{2007}+\ldots+P_{1}\right)=1 / 2008\left(P_{0}-P_{2008}\right)>0$, since

$P_{2008}<1$ (the sum of 2008, obviously, may not occur).

## Answer

$P_{2007}$