Probability
Power Set — P(A∩B=∅), p+q
nta_pyq_2026_jan
Grade 12
Question:
Let $S$ be a set of 5 elements and $P(S)$ denote the power set of $S$. Let $E$ be an event of choosing an ordered pair $(A,B)$ from $P(S)\times P(S)$ such that $A\cap B=\emptyset$. If the probability of the event $E$ is $\dfrac{p}{2^q}$, where $p,q\in\mathbb{N}$, then $p+q$ is equal to _____.
Step-by-Step Solution
Key Concept: Total ordered pairs $=32^2=1024$. For $A\cap B=\emptyset$: each of 5 elements independently belongs to $A$ only, $B$ only, or neither — 3 choices each. Favorable $=3^5=243$.
Probability $=\dfrac{3^5}{2^{10}}$. $p+q=5+10=15$.
Correct Answer: 15