For arbitrary $C>0$ and non-negative i. i. d. $Y_1,\ldots,Y_k$ for $Z:=\sum_{i=1}^k Y_i$ we have $$\sum_{i=1}^k {\mathbf 1}(Z/Y_i\geqslant C)\geqslant k-C,$$and taking the expectation we get $$\mathrm {prob} (Z/Y_1\geqslant C)\geqslant 1-C/k. $$

For arbitrary $C>0$ and positive i. i. d. $Y_1,\ldots,Y_k$ for $Z:=\sum_{i=1}^k Y_i$ we have $$\sum_{i=1}^k {\mathbf 1}(Z/Y_i\geqslant C)\geqslant k-C$$ (since $Z/Y_i<C$ means that $Y_i>Z/C$, which may hold for at most $C$ different values of $i$),

and taking the expectation we get $$\mathrm {prob} (Z/Y_1\geqslant C)\geqslant 1-C/k. $$