Question

The probability of simultaneous occurrence of atleast one of the two events X and Y is a. If the probability that exactly one of the events X, Y occurs is b, prove that \( P(X') + P(Y') = 2 - 2a + b \).

Hint:

"At least one" is the union \( A \cup B \).
"Exactly one" can also be written as \( P(A) + P(B) - 2P(A \cap B) \).

Answer 1

Concept:
• \( P(\text{atleast one}) = P(X \cup Y) = a \).
• \( P(\text{exactly one}) = P(X \cup Y) - P(X \cap Y) = b \).
• Complement rule: \( P(X') = 1 - P(X) \).

Step 1: Find the probability of intersection
From the given information: \[ P(X \cup Y) = a \] \[ P(X \cup Y) - P(X \cap Y) = b \] Substituting \( a \) in the second equation: \[ a - P(X \cap Y) = b \] \[ P(X \cap Y) = a - b \]

Step 2: Express the sum of individual probabilities
Using the addition theorem: \[ P(X \cup Y) = P(X) + P(Y) - P(X \cap Y) \] \[ a = P(X) + P(Y) - (a - b) \] \[ P(X) + P(Y) = a + a - b = 2a - b \]

Step 3: Evaluate the target expression
We need to find \( P(X') + P(Y') \): \[ P(X') + P(Y') = (1 - P(X)) + (1 - P(Y)) \] \[ = 2 - (P(X) + P(Y)) \] Substitute the expression from
Step 2: \[ = 2 - (2a - b) = 2 - 2a + b \] Hence proved.