Find Probability of Event A

Since the odds against event A are $3:5$, the probability of event A is calculated as $P(A) = \frac{5}{3+5} = \frac{5}{8}$ and its complement is $P(A') = \frac{3}{8}$.

$P(A') = \frac{3}{8}$

Find Probability of Event B

Given that the odds in favour of event B are $4:7$, the probability of event B is $P(B) = \frac{4}{4+7} = \frac{4}{11}$ and its complement is $P(B') = \frac{7}{11}$.

$P(B') = \frac{7}{11}$

Apply Independence and Complement Rule

Since A and B are independent, the probability that neither occurs is given by the product of their individual complement probabilities: $\frac{3}{8} \times \frac{7}{11} = \frac{21}{88}$.

$P(A' \cap B') = \frac{21}{88}$

Calculate Probability of At Least One Event

The probability that at least one event occurs is the complement of neither occurring, which is $1 - \frac{21}{88} = \frac{67}{88}$.

$P(A \cup B) = 1 - \frac{21}{88} = \frac{67}{88}$