To solve the problem, we need to find the probability of event Y given that the probability of either event X or event Y occurring is 0.7, and the probability of event X occurring is 0.4. Additionally, we know that events X and Y are independent. ### Step-by-Step Explanation 1. **Understanding the Problem**: - We are given: - \( P(X \cup Y) = 0.7 \) (the probability of either X or Y occurring) - \( P(X) = 0.4 \) (the probability of X occurring) - We need to find \( P(Y) \) (the probability of Y occurring). 2. **Using the Formula for Independent Events**: - For two independent events, the probability of either event occurring can be calculated using the formula: \[ P(X \cup Y) = P(X) + P(Y) - P(X \cap Y) \] - Since X and Y are independent, we can express \( P(X \cap Y) \) as: \[ P(X \cap Y) = P(X) \cdot P(Y) \] 3. **Substituting Known Values**: - We can substitute the known values into the formula: \[ P(X \cup Y) = P(X) + P(Y) - P(X) \cdot P(Y) \] - Plugging in the values we have: \[ 0.7 = 0.4 + P(Y) - (0.4 \cdot P(Y)) \] 4. **Rearranging the Equation**: - Let's simplify the equation: \[ 0.7 = 0.4 + P(Y) - 0.4P(Y) \] - Combine the terms involving \( P(Y) \): \[ 0.7 = 0.4 + P(Y)(1 - 0.4) \] - This simplifies to: \[ 0.7 = 0.4 + 0.6P(Y) \] 5. **Isolating \( P(Y) \)**: - Now, we can isolate \( P(Y) \): \[ 0.7 - 0.4 = 0.6P(Y) \] \[ 0.3 = 0.6P(Y) \] - Dividing both sides by 0.6 gives: \[ P(Y) = \frac{0.3}{0.6} = 0.5 \] ### Conclusion The probability of event Y occurring is \( P(Y) = 0.5 \). ### Evaluating the Options - **Option A: 0.30** - Incorrect. This does not match our calculated probability. - **Option B: 0.50** - Correct. This matches our calculated probability of Y. - **Option C: 0.57** - Incorrect. This does not match our calculated probability. - **Option D: 1.80** - Incorrect. Probabilities cannot exceed 1. ### Revision Summary - The probability of either event X or Y occurring is calculated using the formula for independent events. - For independent events, \( P(X \cap Y) = P(X) \cdot P(Y) \). - Rearranging the equation allows us to isolate and solve for \( P(Y) \). - The final answer is \( P(Y) = 0.5 \).