17. On a roulette wheel, any number from 0 to 2007 can come up with equal probability. The roulette wheel is spun repeatedly. Let $P_{k}$ be 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 $P_{2008} ?$
# Solution.

Reasoning in the same way as in the previous problem (the only difference is that instead of the number 6 - the number 2008), we find:

$$
P_{n}=\frac{1}{2008} \sum_{k=0}^{2007} P_{n-k} ; \frac{2007}{2008} P_{n}=\frac{1}{2008} \sum_{k=1}^{2007} P_{n-k} ; P_{n}=\frac{1}{2007} \sum_{k=1}^{2007} P_{n-k}
$$

Since the sum at the beginning of the game is certainly zero, then $P_{0}=1$. Then

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

since $P_{2007}P_{2008}$.

Answer: $P_{2007}$.