1. **Problem:** Find the probability that either Jack or Bill will win the game, given that the probability Jack wins is $\frac{1}{5}$ and the probability Bill wins is $\frac{1}{4}$. Only one player can win. 2. **Formula:** For two mutually exclusive events $A$ and $B$, the probability that $A$ or $B$ occurs is $$\Pr(A \cup B) = \Pr(A) + \Pr(B)$$ Since only one player can win, the events "Jack wins" and "Bill wins" are mutually exclusive. 3. **Calculation:** $$\Pr(\text{Jack or Bill wins}) = \Pr(\text{Jack wins}) + \Pr(\text{Bill wins}) = \frac{1}{5} + \frac{1}{4}$$ 4. **Simplify:** Find common denominator 20: $$\frac{1}{5} = \frac{4}{20}, \quad \frac{1}{4} = \frac{5}{20}$$ So, $$\Pr(\text{Jack or Bill wins}) = \frac{4}{20} + \frac{5}{20} = \frac{9}{20}$$ 5. **Interpretation:** The probability that either Jack or Bill wins is $\frac{9}{20}$, meaning there is a 45% chance one of them wins. **Final answer:** $$\boxed{\Pr(\text{Jack or Bill wins}) = \frac{9}{20}}$$