Question

A set has n elements. A subset P of A is selected at random. All the elements of P are returned to A. The subset Q of A is formed. Find the probability that P and Q have no common element.

Answer 1

The subset P, formed first, may contain none, one, two,… or all elements of A. P and Q have no common elements. Any element x of A has one of the four possibilities.

1. x ∈ P; x ∈ Q

2. x ∈ P; x ∉ Q

3. x ∉ P; x ∈ Q

4. x ∉ P; x ∉ Q

Case 2, 3, 4 correspond to P and Q having no common element. Hence, the probability that this element x is not in P or not in Q is 3/4. The same is true of the other elements.

Hence, the probability is

nCn(3/4)n(1/4)0