1. The problem asks to find the probability that a randomly chosen integer from 1 to 500 is divisible by 7 or 11. 2. We use the formula for the probability of divisibility by either of two numbers: $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$ where: - $A$ is divisible by 7 - $B$ is divisible by 11 3. Count integers divisible by 7 from 1 to 500: $$\left\lfloor \frac{500}{7} \right\rfloor = 71$$ 4. Count integers divisible by 11 from 1 to 500: $$\left\lfloor \frac{500}{11} \right\rfloor = 45$$ 5. Count integers divisible by both 7 and 11, i.e., divisible by their least common multiple (LCM=77): $$\left\lfloor \frac{500}{77} \right\rfloor = 6$$ 6. Calculate the probability: $$P = \frac{71}{500} + \frac{45}{500} - \frac{6}{500} = \frac{71 + 45 - 6}{500} = \frac{110}{500} = \frac{11}{50}$$ 7. Therefore, the probability is $\boxed{\frac{11}{50}}$.