For any two events A and B, which of the following are correct ?
(A) \( \mathrm{P}(A\cup B)=\mathrm{P}(A)+\mathrm{P}(B)-\mathrm{P}(A\cap B) \)
(B) \( \mathrm{P}(A)=\mathrm{P}(A\cup B)+\mathrm{P}(A\cap B) \)
(C) \( \mathrm{P}(A/B)=\ \mathrm{P}(A)\ /\ \mathrm{P}(B) \)
(D) \( \mathrm{P}(A)=\mathrm{P}(A\cap B)+\mathrm{P}(A\cap\bar{B}) \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (D) only
For any two events A and B, the addition theorem and the definition of conditional probability give
\( P(A\cup B)=P(A)+P(B)-P(A\cap B) \)
\( \displaystyle P(A/B)=\frac{P(A\cap B)}{P(B)} \)
Also, A splits into the disjoint parts \( A\cap B \) and \( A\cap\bar{B} \).
So \( P(A)+P(B)=P(A\cup B)+P(A\cap B) \), \( P(A/B)=\frac{P(A\cap B)}{P(B)} \) and \( P(A)=P(A\cap B)+P(A\cap\bar{B}) \).
Checking the statements
(A) \( \mathrm{P}(A\cup B)=\mathrm{P}(A)+\mathrm{P}(B)-\mathrm{P}(A\cap B) \) — Correct. This is the addition theorem.
(B) \( \mathrm{P}(A)=\mathrm{P}(A\cup B)+\mathrm{P}(A\cap B) \) — Incorrect. The true relation has \( P(A)+P(B) \) on the left, not \( P(A) \).
(C) \( \mathrm{P}(A/B)=\ \mathrm{P}(A)\ /\ \mathrm{P}(B) \) — Incorrect. The numerator should be \( P(A\cap B) \), not \( P(A) \).
(D) \( \mathrm{P}(A)=\mathrm{P}(A\cap B)+\mathrm{P}(A\cap\bar{B}) \) — Correct. A is the union of the disjoint parts \( A\cap B \) and \( A\cap\bar{B} \).