Probability from Venn diagrams — set notation

The notation $n(A)$ asks for HOW MANY elements are in $A$ — a single number — not a list of them. Writing $\{4,7\}$ when $n(A\cap B)$ is asked scores nothing; the answer would be $2$. Read $n(\ldots)$ as "the number of elements in".

A circle total counts everyone in that set, INCLUDING those in the overlap. To fill the "only $A$" region, subtract the intersection first, so the count is $n(A)-n(A\cap B)$. Writing the full circle total in the "only" region double-counts the people who are in both.

The universal set $\xi$ also contains elements in NEITHER set — the region outside both circles. Forgetting it means the regions do not add up to $n(\xi)$. Always work it out: outside $= n(\xi) - (\text{only }A + \text{both} + \text{only }B)$.

On set-notation questions, not shading the region the notation describes leads to picking the wrong parts. Shade exactly what $A\cap B$, $A\cup B$ or $A'$ means FIRST, then read the numbers from the shaded region only. Guessing which regions to combine is the classic error.

For a union count, use $n(A\cup B)=n(A)+n(B)-n(A\cap B)$. Adding $n(A)+n(B)$ alone counts everyone in the overlap TWICE, giving too large a total. Subtract the intersection once. E.g. $n(A)=18$, $n(B)=14$, $n(A\cap B)=5$ gives $18+14-5=27$, not $32$.

When a Venn question asks for a PROBABILITY, divide the region count by the universal-set total: $P(A)=\dfrac{n(A)}{n(\xi)}$. Leaving the answer as a raw count (e.g. $3$ instead of $\dfrac{3}{30}=0.1$) loses the mark. A probability is a number from $0$ to $1$.