1. **State the problem:** We need to find the probability that Arun scores the winning goal on a Saturday. 2. **Given information:** - Probability Arun plays football on a Saturday: $P(F) = \frac{3}{4}$ - Probability Arun scores the winning goal given he plays football: $P(S|F) = \frac{1}{7}$ 3. **Formula used:** The probability that Arun scores the winning goal on a Saturday is the product of the probability he plays football and the probability he scores given he plays. This is given by the multiplication rule for dependent events: $$P(S) = P(F) \times P(S|F)$$ 4. **Calculate:** $$P(S) = \frac{3}{4} \times \frac{1}{7} = \frac{3}{28}$$ 5. **Interpretation:** The probability that Arun scores the winning goal on a Saturday is $\frac{3}{28}$, which means out of 28 Saturdays, on average, Arun scores the winning goal 3 times. **Final answer:** $$\boxed{\frac{3}{28}}$$