Does the time when you buy a sweepstake ticket change your odds of winning?

Context:

At work there is a Grand National (horse racing) Sweepstake Competition.

Rules of the Sweepstake:

there are 40 tickets, each ticket is assigned to a horse (not assigned yet), you can buy maximum of 2 tickets, all tickets must be sold, who ever holds the ticket with the number assigned to the winning horse wins

My Question:

Based on the rules of the sweepstake described above if i wait for a certain number of tickets to be sold before buying my ticket does this affect my odds of winning? if so when would be the optimum time to pick a ticket i.e. after how many tickets sold should I buy my ticket to have the best chance of winning

My thinking:

I assume it does because if there was 100 tickets 99 losers and 1 winner and some one else buys one then statistically the chances of them picking the winning ticket is low (1/100) so if you pick next chances are, your now picking out 98 losing tickets and 1 winner so your chances are now 1/99 (minus the chance of the 1st person picking the winning ticket) I know that this sort of basic intuitive thinking is usually wrong when it come to statistics and probability.

$$p_w=\frac{1}{N-M}\cdot\frac{N-M}{N}=\frac{1}{N}$$
with $$N$$ being the overall number of tickets and $$M$$ being the number of tickets already drawn. $$\frac{N-M}{N}$$ is the chance, that the winning ticket has not yet been sold.