Q.A die is rolled once. Find the probability of getting an even number or a number greater than 4, using the addition theorem.

Define the two events, find their individual and overlap probabilities, then apply the general addition theorem since the events overlap.

Step 1 — Define the events. Let $A$ = 'an even number' $=\{2,4,6\}$, so $n(A)=3$, $P(A)=3/6$. Let $B$ = 'a number greater than 4' $=\{5,6\}$, so $n(B)=2$, $P(B)=2/6$.

Step 2 — Find the overlap. $A\cap B$ = numbers that are both even AND greater than 4 $=\{6\}$, so $n(A\cap B)=1$, $P(A\cap B)=1/6$. Since the two events share the outcome $6$, they are not mutually exclusive, so the general form of the addition theorem must be used.

Step 3 — Apply the addition theorem.

$P(A\cup B) = P(A)+P(B)-P(A\cap B) = \dfrac{3}{6}+\dfrac{2}{6}-\dfrac{1}{6} = \dfrac{4}{6} = \dfrac{2}{3}$ …

Listing every outcome directly, as in the Independent check, is always a reliable cross-check for a small, enumerable sample space like a single die roll, and …

Adding $P(A)+P(B)$ directly without subtracting the overlap (getting $5/6$ instead of $2/3$) is the classic error whenever two events are wrongly assumed to be mutually exclusive — always che …