This problem involves calculating the probability of the union of two independent random events, X and Y, given some initial probabilities. We are provided with \( P(X) \) and \( P(X \cup Y^C) \), where \( Y^C \) represents the complement of event Y. Our goal is to determine the value of \( P(X \cup Y) \).

Understanding Independent Random Events

In probability theory, two events are considered independent events if the occurrence of one does not affect the probability of the other. For independent events X and Y:

• The probability of both events occurring (their intersection) is the product of their individual probabilities: \( P(X \cap Y) = P(X) \cdot P(Y) \).
• Similarly, if X and Y are independent, then X and \( Y^C \) (the complement of Y) are also independent. This means \( P(X \cap Y^C) = P(X) \cdot P(Y^C) \).
• The complement of an event \( Y^C \) is the event that Y does not occur. Its probability is \( P(Y^C) = 1 - P(Y) \).
• The union of two events, \( P(A \cup B) \), is the probability that at least one of the events A or B occurs. The general formula for the union of any two events A and B is: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
• If A and B are independent, this formula simplifies to: \[ P(A \cup B) = P(A) + P(B) - P(A) \cdot P(B) \]

Given Probabilities and Goal

We are given the following probabilities:

Our objective is to find the value of \( P(X \cup Y) \).

Deriving P(Y) from P(X ∪ Y^C)

To find \( P(X \cup Y) \), we first need to determine \( P(Y) \). We can use the given probability \( P(X \cup Y^C) \) and the properties of independent events.

We know the general formula for the union of two events A and B is: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Applying this to \( X \) and \( Y^C \): \[ P(X \cup Y^C) = P(X) + P(Y^C) - P(X \cap Y^C) \] Since events X and Y are independent, events X and \( Y^C \) are also independent. Therefore, the probability of their intersection is the product of their individual probabilities: \[ P(X \cap Y^C) = P(X) \cdot P(Y^C) \] Substitute this into the union formula: \[ P(X \cup Y^C) = P(X) + P(Y^C) - P(X) \cdot P(Y^C) \] Now, we can substitute the given values into this equation: \[ 0.7 = 0.4 + P(Y^C) - (0.4 \cdot P(Y^C)) \] Combine the terms involving \( P(Y^C) \): \[ 0.7 = 0.4 + P(Y^C) (1 - 0.4) \] \[ 0.7 = 0.4 + P(Y^C) (0.6) \] Subtract 0.4 from both sides: \[ 0.7 - 0.4 = P(Y^C) (0.6) \] \[ 0.3 = P(Y^C) (0.6) \] To find \( P(Y^C) \), divide 0.3 by 0.6: \[ P(Y^C) = \frac{0.3}{0.6} = 0.5 \] Finally, we can find \( P(Y) \) using the relationship \( P(Y^C) = 1 - P(Y) \): \[ 0.5 = 1 - P(Y) \] \[ P(Y) = 1 - 0.5 = 0.5 \] So, the probability of event Y is 0.5.

Calculating P(X ∪ Y)

Now that we have \( P(X) = 0.4 \) and \( P(Y) = 0.5 \), and we know X and Y are independent events, we can calculate \( P(X \cup Y) \) using the simplified union formula for independent events:

Final Probability Value

The calculated value of \( P(X \cup Y) \) is 0.7.