(D) Q is a subset of P is

For each element $a \in A$, we have four possible states: (1) in neither P nor Q , (2) in P only, (3) in Q only, (4) in both P and Q . For $Q \subseteq P$, an element cannot be in Q only. So each element has only 3 choices: not in P or Q , in P only, or in both P and Q . Number of favorable outcomes: $3^n$ Probability: $\frac{3^n}{4^n} = \left(\frac{3}{4}\right)^n$

Probability

Independent Events and Subset Selection

Grade 12

Question:

(D) Q is a subset of P is

(P) $\frac{\binom{2n}{n}}{4^n}$

(Q) $\frac{2^{2n} - \binom{2n}{n}}{2^{2n+1}}$

(R) $\frac{\binom{2n}{n+1}}{4^n}$

(S) $\left(\frac{3}{4}\right)^n$

(T) $\frac{\binom{2n}{n}}{4^{n-1}}$

Step-by-Step Solution

Key Concept: For subset inclusion, each element has exactly 3 valid states (cannot be in Q without being in P).

For each element $a \in A$, we have four possible states: (1) in neither P nor Q, (2) in P only, (3) in Q only, (4) in both P and Q.For $Q \subseteq P$, an element cannot be in Q only. So each element has only 3 choices: not in P or Q, in P only, or in both P and Q.Number of favorable outcomes: $3^n$Probability: $\frac{3^n}{4^n} = \left(\frac{3}{4}\right)^n$

Correct Answer: S

Question:

Step-by-Step Solution

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