The odds against a certain event A are 4:7 and the odds in favor of another independent event B are 3:4. What is the probability that at least one of the events will occur?

To find the probability that at least one of the events will occur, we can use the concept of odds and convert them into probabilities.

Step 1: Convert the odds against event A into a probability. The odds against event A are given as 4:7. To convert this into a probability, we divide the number of unfavorable outcomes (4) by the total number of outcomes (4 + 7 = 11). So, the probability of event A not occurring is 4/11.

Step 2: Convert the odds in favor of event B into a probability. The odds in favor of event B are given as 3:4. To convert this into a probability, we divide the number of favorable outcomes (3) by the total number of outcomes (3 + 4 = 7). So, the probability of event B occurring is 3/7.

Step 3: Find the probability that at least one of the events will occur. We can use the complement rule, which states that the probability of the complement of an event is equal to 1 minus the probability of the event not occurring.

The probability of neither event A nor event B occurring is the product of their individual probabilities: (4/11) * (4/7) = 16/77.

Therefore, the probability that at least one of the events will occur is 1 - (16/77) = 61/77.

So, the probability that at least one of the events will occur is 61/77.