Addition Rule: P(A or B) with Overlapping Events

The general addition rule

For any two events $A$ and $B$ in a sample space: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

The subtraction fixes double-counting: any outcome in both $A$ and $B$ was counted once in $P(A)$ and once in $P(B)$, so it must be removed once.

When $A$ and $B$ are mutually exclusive, $P(A \cap B) = 0$ and the rule collapses to the simpler version $P(A \cup B) = P(A) + P(B)$.

Step-by-step solution

We want $P(\text{Heart OR Face card})$ from a standard 52-card deck. "Face card" means Jack, Queen, or King.

Step 1 — $P(\text{Heart})$. 13 hearts in 52 cards: $P(H) = \dfrac{13}{52} = \dfrac{1}{4}$

Step 2 — $P(\text{Face})$. J, Q, K in each of 4 suits = 12 face cards: $P(F) = \dfrac{12}{52} = \dfrac{3}{13}$

Step 3 — $P(\text{Heart AND Face})$. The heart-face cards are J♥, Q♥, K♥ — three cards: $P(H \cap F) = \dfrac{3}{52}$

Step 4 — Apply the addition rule: $P(H \cup F) = \tfrac{13}{52} + \tfrac{12}{52} - \tfrac{3}{52} = \boxed{\tfrac{22}{52} = \tfrac{11}{26}}$

That's roughly 42.3%.

Verification by direct count

Which cards are in $H \cup F$? All 13 hearts, plus the face cards that aren't hearts (J♠, Q♠, K♠, J♦, Q♦, K♦, J♣, Q♣, K♣ = 9 cards). Total favorable: $13 + 9 = 22$. Probability: $22/52 = 11/26$. ✓

Visual intuition — Venn diagrams

Picture two overlapping circles: "Hearts" and "Face cards." Their union is the full shaded region. If we added the circle areas directly, the lens-shaped overlap in the middle would be counted twice — once from each circle. Subtracting $P(H \cap F)$ (the overlap) once brings the count back to exactly one per card.

Extending to three events (inclusion–exclusion)

$P(A \cup B \cup C) = P(A) + P(B) + P(C)$ $\quad - P(A \cap B) - P(A \cap C) - P(B \cap C)$ $\quad + P(A \cap B \cap C)$

Alternating signs: add singles, subtract pairs, add triples.

Common mistakes

• Skipping the subtraction. $P(H) + P(F) = 25/52$ — larger than the true answer of $22/52$. The extra $3/52$ is the double-counted overlap.
• Misidentifying the overlap. Think carefully: which outcomes belong to both events? Enumerate if needed.
• Assuming events are disjoint when they are not. Always ask, "Can a single outcome satisfy both descriptions?" If yes, the events overlap.

Try it in the visualization

The Venn diagram updates live as you tweak $P(A)$, $P(B)$, and $P(A \cap B)$. Any invalid combination (like overlap bigger than either event) is flagged. A highlighted strip at the bottom shows the three pieces $P(A) + P(B) - P(A \cap B)$ on a number line.