Probability: Diagram-Based Questions (7)

Question

In a Venn diagram, set $A$ contains 60 elements, set $B$ contains 50 elements, and their intersection contains 20 elements. The total sample space has 100 elements. A point is chosen at random. Find: (i) $P(A)$, (ii) $P(B)$, (iii) $P(A \cup B)$, (iv) $P(A \cap B^c)$, (v) $P(A^c \cap B^c)$.

Solution — Step by Step

Final answers: (i) $0.6$ (ii) $0.5$ (iii) $0.9$ (iv) $0.4$ (v) $0.1$.

Why This Works

A Venn diagram partitions the sample space into four mutually exclusive regions: $A$ only, $B$ only, both, and neither. Each region’s probability adds to $1$. Once you know any three, you know all four.

The inclusion-exclusion formula $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ is just a restatement that you should not double-count the intersection.

Alternative Method

Build a 2×2 contingency table with rows “in $A$” / “not in $A$” and columns “in $B$” / “not in $B$”. Fill in the four cells from given totals. Read off any probability directly. This is faster for problems with three or more sets.