If U = x : x is an integer and 1 ≤ x ≤ 20 E1 = x: x is a multiple of 3 E2 = x: x is a multiple of 4 and an…

Analyze the question: We need the probability that a number picked from U = {1, 2, ..., 20} is NOT a multiple of 4.

Concept: P(not E2) = 1 − P(E2), or equivalently (number of elements not in E2) / (total number of elements).

Solution:

1. Total number of elements in U: n(U) = 20.
2. E2 = multiples of 4 from 1 to 20 = {4, 8, 12, 16, 20}, so n(E2) = 5.
3. P(E2) = 5/20 = 1/4.
4. P(not in E2) = 1 − 1/4 = 3/4.

Check: the numbers not in E2 number 20 − 5 = 15, and 15/20 = 3/4. ✓

Tip: When a question says "not in", use the complement rule: P(A′) = 1 − P(A). It is usually faster than counting the elements outside the set. Also, to count multiples of k up to n, just divide: 20 ÷ 4 = 5.