Addition Rule of Probability

The addition rule tells you the probability that at least one of two events occurs — in other words, $P(A \text{ or } B)$. This is one of the most important rules in probability, and getting it right depends on understanding whether the two events can happen at the same time.

There are two versions of the addition rule: a simple one for mutually exclusive events and a general one for overlapping events. This page covers both, shows you how to visualize them with Venn diagrams, and walks through the common mistakes students make.

Mutually Exclusive Events

Two events are mutually exclusive (also called disjoint) if they cannot happen at the same time. When one occurs, the other is impossible.

Examples of mutually exclusive events:

• Rolling a 2 or rolling a 5 on a single die — you cannot roll both at once
• A student being in grade 9 or grade 10 — you cannot be in both grades simultaneously
• Drawing a heart or drawing a club from a deck — a single card cannot be both suits

For mutually exclusive events, $P(A \text{ and } B) = 0$. There is zero overlap, so the addition rule simplifies to:

$P(A \text{ or } B) = P(A) + P(B)$

Example 1: Rolling a Die

What is the probability of rolling a 2 or a 5 on a standard six-sided die?

These events are mutually exclusive — a single roll cannot be both 2 and 5.

$P(2 \text{ or } 5) = P(2) + P(5) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$

Answer: The probability is $\frac{1}{3}$, or approximately 0.333.

The General Addition Rule

When two events can overlap — meaning they can happen at the same time — you must use the general addition rule:

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

Why subtract the overlap? If you simply add $P(A)$ and $P(B)$, you count every outcome in the overlap twice — once when counting $A$ and again when counting $B$. Subtracting $P(A \text{ and } B)$ corrects for this double-counting.

Notice that the general rule works for mutually exclusive events too: when $P(A \text{ and } B) = 0$, the formula reduces to $P(A) + P(B)$.

Example 2: Cards — Red or Face Card

What is the probability of drawing a red card or a face card from a standard 52-card deck?

These events overlap because some face cards are red (the jack, queen, and king of hearts and diamonds).

Identify the counts:

• Red cards: 26 (all hearts and diamonds)
• Face cards: 12 (jack, queen, king in each of 4 suits)
• Red face cards (overlap): 6 (jack, queen, king of hearts + jack, queen, king of diamonds)

Apply the general addition rule:

$P(\text{red or face}) = \frac{26}{52} + \frac{12}{52} - \frac{6}{52} = \frac{32}{52} = \frac{8}{13}$

Answer: The probability is $\frac{8}{13} \approx 0.615$, or about 61.5%.

Verification: The 32 favorable cards break down as: 20 red non-face cards + 6 red face cards + 6 non-red face cards = 32. And $52 - 32 = 20$ cards are neither red nor face cards (the non-face black cards: ace through 10 of spades and clubs). That checks out: $10 \times 2 = 20$.

Venn Diagrams for Probability

A Venn diagram is a visual tool for understanding how events overlap. Two overlapping circles represent events $A$ and $B$, placed inside a rectangle that represents the entire sample space.

Here is the Venn diagram for the card example above:

Reading the Venn diagram: the left circle contains all 26 red cards (20 that are not face cards plus 6 that are). The right circle contains all 12 face cards (6 that are not red plus 6 that are). The overlap holds the 6 red face cards. The 20 cards outside both circles are the non-face, non-red cards (ace through 10 of spades and clubs).

Check: $20 + 6 + 6 + 20 = 52$. All 52 cards are accounted for.

Common Mistake: Forgetting to Subtract the Overlap

The most frequent error students make is using the simple addition rule when the events are not mutually exclusive:

$\text{Wrong: } P(\text{red or face}) = \frac{26}{52} + \frac{12}{52} = \frac{38}{52}$

This answer, $\frac{38}{52} \approx 0.731$, is too large because the 6 red face cards have been counted twice. The correct answer subtracts the overlap:

$\text{Right: } P(\text{red or face}) = \frac{26}{52} + \frac{12}{52} - \frac{6}{52} = \frac{32}{52} \approx 0.615$

How to avoid the mistake: Always ask yourself, “Can events $A$ and $B$ happen at the same time?” If the answer is yes — or if you are not sure — use the general addition rule. It works in every case, because when there is no overlap the subtracted term is simply zero.

”At Least One” Problems

A powerful shortcut for “at least one” problems uses the complement rule:

$P(\text{at least one}) = 1 - P(\text{none})$

This works because “at least one” and “none” are complementary events — one of them must occur — and complementary probabilities add to 1.

Example 3: At Least One Head

You flip a fair coin 3 times. What is the probability of getting at least one head?

Step 1: Find $P(\text{no heads})$. “No heads” means all three flips are tails:

$P(\text{no heads}) = \left(\frac{1}{2}\right)^3 = \frac{1}{8}$

Step 2: Subtract from 1.

$P(\text{at least one head}) = 1 - \frac{1}{8} = \frac{7}{8} = 0.875$

Answer: The probability of getting at least one head in 3 flips is $\frac{7}{8}$, or 87.5%.

Why the complement approach is better: The direct approach would require counting all outcomes with 1 head, 2 heads, or 3 heads and adding them up. The complement approach takes one subtraction. As the number of trials increases, the complement shortcut saves enormous effort.

Real-World Application: Nursing — Probability of Patient Conditions

In a hospital ward, patient records show the following prevalence:

• $P(\text{diabetes}) = 0.20$ (20% of patients have diabetes)
• $P(\text{hypertension}) = 0.35$ (35% have hypertension)
• $P(\text{both}) = 0.12$ (12% have both conditions)

What is the probability that a randomly selected patient has diabetes or hypertension (or both)?

Since the two conditions can occur together, use the general addition rule:

$P(\text{diabetes or hypertension}) = 0.20 + 0.35 - 0.12 = 0.43$

Answer: There is a 43% chance that a patient has at least one of these conditions.

Interpretation for care planning: This means that 43% of the ward needs monitoring for at least one of these conditions — but you cannot simply add the two prevalences (which would give 55%) because that double-counts the 12% who have both. The Venn diagram breakdown:

• Diabetes only: $0.20 - 0.12 = 0.08$ (8%)
• Hypertension only: $0.35 - 0.12 = 0.23$ (23%)
• Both: $0.12$ (12%)
• Neither: $1 - 0.43 = 0.57$ (57%)

Verification: $0.08 + 0.23 + 0.12 + 0.57 = 1.00$. All probabilities sum to 1.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• The addition rule finds $P(A \text{ or } B)$ — the probability that at least one of two events occurs
• For mutually exclusive events (no overlap): $P(A \text{ or } B) = P(A) + P(B)$
• For overlapping events: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$
• When in doubt, always use the general rule — it works for both cases
• A Venn diagram makes the overlap visible and helps verify that all outcomes sum to the sample space
• For “at least one” problems, use the complement: $P(\text{at least one}) = 1 - P(\text{none})$
• The most common error is forgetting to subtract the overlap, which double-counts shared outcomes

Return to Statistics for more topics in this section.