<s>[INST] It seems that the payoff of a part of a lottery ticket is roughly proportional to the logarithm of the number on the ticket. If we have a logarithmic scale and the probability of buying the ticket with number N is \( P(N) = 1 - \frac{\log( N + 1 )}{\log( N + 2 )} \), what is the probability of buying a ticket with a number between 1 and \(N\)?[/INST]