What is P(A ⊆ B)?

Let X have n elements. Choose A and B independently and uniformly from all subsets of X.

Look at one element x ∈ X

For each element, membership in A and B gives four equally likely possibilities:

x ∉ A, x ∉ B

✓ allowed

x ∉ A, x ∈ B

✓ allowed

x ∈ A, x ∉ B

✗ forbidden

x ∈ A, x ∈ B

✓ allowed

For A ⊆ B, only one of the four states is forbidden.
P(one element is compatible) = 3/4

Number of elements n: 4

All n elements must avoid the forbidden state.

(3/4)^4 = 81/256

≈ 31.64%

Counting shortcut:

Total ordered pairs (A, B): 4^n

Valid pairs with A ⊆ B: 3^n

P(A ⊆ B) = (3/4)ⁿ

Because each element independently has 3 valid membership states out of 4.