We have : $$
P(S \geq m+1) = P(X_1 + ... + X_m \leq n)
$$
where $X_i$ are iid uniformly distributed in $\{1,2,...,n\}$. Therefore, it is sufficient to find the RHS.
The answer is just $\sum_{1 \leq i_1,...,i_m \leq n} 1_{i_1+...+i_m \leq n} P(X_1 = i_1,X_2 = i_2,...,X_m = i_m)$, which just becomes $\frac {K}{n^m}$, where :
$$
K = |\{(i_1,i_2,...,i_m) \in \{1,2,...,n\}^m : i_1+i_2+...+i_m \leq n\}|
$$
We claim that the set given in the definition of $K$ is in bijection with the set $\{(i_1,i_2,...,i_m) \in \{0,1,...,m\}^m : i_1+i_2+...+i_m \leq n-m\}$. The bijection is given by taking a tuple and subtracting $1$ from each entry. It is easy to see, as $n>m$ that the bijection holds.
Now, we adjoin an index $i_{m+1} = n-m - (i_1+...+i_m)$, so we get that the above set is in bijection with :
$$
\{(i_1,i_2,...,i_m,i_{m+1}) \in \{0,1,...,m\}^{m+1} : i_1+i_2+...+i_m +i_{m+1} = n-m\}
$$
by projection onto the first $m$ coordinates.
This last set is calculated using stars and bars. Indeed, imagine a set of $n$ blanks, each filled with a $|$ or a $\circ$ (ball), so that there are exactly $m$ of $|$ and $n-m$ of $\circ$.
$$
\underbrace{|\circ||\circ\circ\circ|\circ||...|\circ| \circ \circ}_{\text{$n$ blanks}}
$$
The bars separate the circles into $m+1$ parts, each containing $i_1,...,i_{m+1}$ number of circles, which total to $n-m$ circles, and each of which is non-negative. In the example above, $i_1 = 0,i_2 = 1,i_3 =0 , i_4 = 3$ and so on.
It is easy to see that the number of ways of placing these bars and circles is the same as the number of elements in the set mentioned earlier. But we are basically placing $m$ bars into $n$ slots : the answer to this is just $\binom{n}{m}$.
Thus, we basically have $K = \binom{n}{m}$, with $Pr(S \geq m+1) = \frac{\binom{n}{m}}{n^m}$.
The expectation of $S$ is equal to $\sum_{i=0}^n \binom ni n^{-i} = (1 + \frac 1n)^n \to e$ as $n \to \infty$ (i.e. as we approach the uniform distribution)
Best Answer
Choosing two numbers in (0,1) is the same that choosing a point in the square $(0,1)^2$. Draw the half-plane $x+y\ge 1$ and you will see the answer...