With a fixed number of entries, is it better to play at a single sweepstake rather than many? The type of the sweepstake is that there is only one prize ($\$100$) and at each sweepstake one and only one winner is guaranteed. Suppose I have two entries and there are $99$ other entry in Sweepstake $1$ and $99$ other entries in Sweepstake $2$. $Equity_{1play} = 100 \cdot \frac2{101} \approx 1.9801980198$ $Equity_{2play} = 100\cdot\frac1{100}+100\cdot\frac1{100}-200\cdot(\frac1{100})^2=1.98$ The second expression follows the probabilistic axiom $P(A\text{ or }B) = P(A) + P(B) - P(A\text{ and }B)$ Therefore, I think playing once is slightly better than playing many times, unless I'm doing an unstable computation on the calculator or I'm misusing the OR axiom.

I take it there are $101$ tickets overall, since you say that you have two entries and there are $99$ other entries.

Then in version $1$, our expectation is $100 \cdot \frac{2}{101}$.

In version $2$, our expectation is $100 \left(\frac{1}{101}+\frac{1}{101}\right)$: the same.