You are correct!

The calculation: 1/1600 x 300/1600 is not the probability of someone winning.

The calculation as Link supplied them is approaching the problem from the point of view of a random club member, 'playing against the all other members' ie what is the probability that any random club member gets his name drawn and has signed in. I think your previous wording confuses people as to what you are really after.

Is this then what you mean?

Calculating the probability of someone winning [that is anyone on a given Monday], is better explained from the point of view of the club; what is the chance that the club will give away a prize on a Monday - and yes, this is strongly dependent on the amount of 'playing members' ie the book signers.

The chance depends on the participants: If nobody signs, its 0 and if everyone signs its 1. The probability is also additive, each of the individual book signers on Monday has a 1/1600 chance of wining but 'the probability of someone winning' is the combined probability of them all: 300/1600)

It can then be calculated as: (1/1600) x N booksigners

When 300 people sign-in, its already high: 300/1600 (and of course when everyone signs its 1/1600 x 1600 = 1600/1600: they will have to give away a prize).