Probability of Winning Per Day

if you enter multiple sweepstakes changes the probability

Probability to win? Yes, obviously the chance to win gets higher. Probability to win a specific giveaway? No, it's independent from the others.
The probability of winning a specific giveaway is 1/[number of entries in that giveaway].
The probability of winning in a day is 1 - [chance of not winning anything] = 1- ([number of entires in giveaway 1 -1]/[[number of entires in giveaway 1])([number of entires in giveaway 2 -1]/[number of entires in giveaway 2])....([number of entires in giveaway n -1]/[[number of entires in giveaway n]) where n is the number of giveaways you have entered in which end during this day.

I assume it is done mathematical correct, so yes, entering multiple giveaways changes the odds.

If you enter one giveaway and there are 9 users beside you the probability to win is 10%.

If you enter two giveaways with 10% each your chance to win at least one of them is

1-0.9*0.9 = 0.19

or 19%.

With more realistic number: You enter 100 giveaways with 1000 entries each. You chance to win at least one is

1-0.999^100 = 0.095

or 9.5%.

Hope that helps.

I can tell you from multiple experiences with entering multiple giveaways with very low numbers of entrants at the same time, that Oppenh4imer is correct.
I once took part in 5 giveaways with each less than 10 participants, and figured it out; Another time, I was one of the first to solve a rather difficult puzzle event, including being among the first to enter the top prizes, and I watched the way my win probability changed every day as new people entered. More recently, I was one of only two people to enter a particular giveaway, which let me confirm this is how it worked

edit: keep in mind that the graph rounds to whole numbers

For each giveaway you enter, the probability of you winning is the number of copies offered divided by the total number of entries. Call this x.
The probability of you NOT winning is 1-x. Call this y.

The probability you will win at least something over some set of giveaways is 1 minus the product of all y for those giveaways.
The expected number of wins over some set of giveaways is the sum of all x for those giveaways.

Notice that if you are definitely going to win a certain giveaway, then x for that giveaway is 1, and y for that giveaway is 0. Thus the product of all y for that day will be 0 and so the probability that you win something that day will still be 100% regardless of the other giveaways you enter.

Also notice that y is always a fraction, always less than 1. Thus, multiplying anything by y will give a LOWER value. So the probability you win at least one thing in a day can only increase by entering more giveaways. Entering a giveaway can never harm your chances.