Two subsets $A$ and $B$ of a set containing $n$ elements are chosen at random. The probability that $A \subseteq B$ is:
Step-by-Step Solution
Key Concept: For each element in the set, count independent choices: element can be in neither A nor B, in B only, or in both A and B (3 choices per element). Since A ⊆ B is satisfied in exactly 3 out of 4 equally likely (A,B) configurations per element, the probability is (3/4)^n.
The favorable outcomes are all subsets containing at least one element: $\binom{n}{1}2^0 + \binom{n}{2}2^1 + \cdots + \binom{n}{n}2^0 = 3^n$. Total outcomes are $2^n \times 2^n = 4^n$, so the required probability is $\left(\frac{3}{4}\right)^n$.
Correct Answer: 4