Question

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.

• A. 7/18
• B. 7/11
• C. 5/8
• D. 8/19

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, \( P(A \cap B) = P(A) \times P(B) \).

Answer 1

The Correct Option is A

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.