If the odds in favour of an event A are 3 : 4 and the odds against another independent event B are 7 : 4, find the probability that at least one of the events will happen.
Show Hint
For independent events, use the formula \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \), and remember that for independent events, \( P(A \cap B) = P(A) \times P(B) \).
Step 1: Understanding the odds.
The odds in favour of event A are 3:4. This means the probability of event A occurring is given by:
\[
P(A) = \frac{3}{3 + 4} = \frac{3}{7}
\]
The odds against event B are 7:4, meaning the probability of event B occurring is:
\[
P(B) = \frac{4}{7 + 4} = \frac{4}{11}
\]
Step 2: Finding the probability of at least one event occurring.
The probability that at least one of the events A or B occurs is given by the formula:
\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]
Since events A and B are independent, the probability of both events A and B occurring is:
\[
P(A \cap B) = P(A) \times P(B) = \frac{3}{7} \times \frac{4}{11} = \frac{12}{77}
\]
Step 3: Calculating the final probability.
Now, substitute the values into the formula for the union of the events:
\[
P(A \cup B) = \frac{3}{7} + \frac{4}{11} - \frac{12}{77}
\]
To simplify, find a common denominator, which is 77:
\[
P(A \cup B) = \frac{33}{77} + \frac{28}{77} - \frac{12}{77} = \frac{49}{77} = \frac{7}{18}
\]
Step 4: Conclusion.
The correct answer is (A) 7/18.
