1. The problem asks for the probability that an integer chosen from the set $\{1, 2, 3, \ldots, 500\}$ is divisible by 7 or 11. 2. Use the formula for the probability of the union of two events: $$P(7 \text{ or } 11) = P(7) + P(11) - P(7 \text{ and } 11)$$ 3. Calculate the number of integers divisible by 7: $$\left\lfloor \frac{500}{7} \right\rfloor = 71$$ 4. Calculate the number of integers divisible by 11: $$\left\lfloor \frac{500}{11} \right\rfloor = 45$$ 5. Calculate the number of integers divisible by both 7 and 11 (i.e., divisible by $7 \times 11 = 77$): $$\left\lfloor \frac{500}{77} \right\rfloor = 6$$ 6. Calculate the total count divisible by 7 or 11: $$71 + 45 - 6 = 110$$ 7. The total number of integers in the set is 500, so the probability is: $$\frac{110}{500} = \frac{11}{50}$$ 8. Therefore, the probability that a randomly chosen integer from 1 to 500 is divisible by 7 or 11 is $\boxed{\frac{11}{50}}$.