The General Addition Rule

The General Addition Rule | CodeSignal Learn

Introduction

P(A \text{ or } B)

From Double-Counting to a Fix

P(A) + P(B)

The General Addition Rule

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

Applying the Rule Step by Step: Part 1

A

Applying the Rule Step by Step: Part 2

61.5\%

A Business Example

A

The Mutually Exclusive Case Revisited

P(A \text{ and } B) = 0

What About More Than Two Events?

A

Common Mistakes to Watch For

P(A \text{ and } B)

Conclusion and Next Steps

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

Previous Lesson

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Recall that adding $P(A) + P(B)$ when events share outcomes always overshoots the true probability. The overshoot equals exactly $P(A \text{ and } B)$ , because every shared outcome gets counted once inside $P(A)$ and then again inside $P(B)$ .

The fix is refreshingly simple: after adding, subtract the overlap once. That removes the extra copy of every shared outcome and leaves each outcome counted exactly once. Think of it like merging two contact lists on your phone — you combine both lists and then remove the duplicates so no one appears twice.

Here is the formula that puts this fix into practice:

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

Each piece has a clear job:

$P(A)$ captures every outcome in event $A$ .
$P(B)$ captures every outcome in event $B$ .
$P(A \text{ and } B)$ is the overlap — the outcomes counted in both of the first two terms.

Subtracting $P(A \text{ and } B)$ removes the second copy of those shared outcomes so the final result reflects each outcome only once. This formula works for any two events, overlapping or not, which is why it earns the name "general."

Venn diagram showing events A and B with labeled regions for P(A) only, P(B) only, and the overlap P(A and B), illustrating how the general addition rule maps to each region

Let's revisit the playing-card scenario from the last lesson and walk through the formula carefully. We draw one card from a standard 52-card deck. Event $A$ is that the card is red (26 cards), event $B$ is that the card is a face card (12 cards), and the overlap $A \text{ and } B$ is that the card is a red face card (6 cards).

With all three values identified, we apply the general addition rule one term at a time:

Find $P(A)$ : There are 26 red cards out of 52, so $P(A) = \frac{26}{52}$ .
Find $P(B)$ : There are 12 face cards out of 52, so $P(B) = \frac{12}{52}$ .
Find $P(A \text{ and } B)$ : There are 6 red face cards, so $P(A \text{ and } B) = \frac{6}{52}$ .
Combine:

P(A \text{ or } B) = \frac{26}{52} + \frac{12}{52} - \frac{6}{52} = \frac{32}{52} \approx 0.615

The probability of drawing a card that is red or a face card (or both) is about $61.5\%$ . Subtracting $\frac{6}{52}$ brought us from the inflated $\frac{38}{52}$ we identified last lesson down to the correct $\frac{32}{52}$ .

Venn diagram of the playing card example showing counts of 20 red non-face cards, 6 red face cards in the overlap, and 6 non-red face cards, with the general addition rule calculation displayed below

Now let's see the formula at work in a real-world business setting. An online retailer analyzes 400 customer accounts to see which product categories each person shopped from last month:

Category	Customers
Electronics	140
Clothing	100
Both categories	40

We want the probability that a randomly selected customer bought from at least one of these two categories. Let $A$ be "purchased electronics" and $B$ be "purchased clothing."

P(A) = \frac{140}{400} = 0.35, \quad P(B) = \frac{100}{400} = 0.25, \quad P(A \text{ and } B) = \frac{40}{400} = 0.10

Plugging into the formula:

P(A \text{ or } B) = 0.35 + 0.25 - 0.10 = 0.50

Half of the customer base bought electronics or clothing (or both) last month. Without subtracting the overlap, we would have reported $0.60$ — overstating the result by $10$ percentage points. In a marketing or inventory decision, that kind of error could lead to seriously misallocated budget.

You might wonder how this new formula relates to the simpler addition rule from Lesson 2. When two events are mutually exclusive, they share no outcomes, which means $P(A \text{ and } B) = 0$ . Substituting that into the general formula gives:

P(A \text{ or } B) = P(A) + P(B) - 0 = P(A) + P(B)

That is exactly the rule you already learned! The simpler formula is just a special case of the general one. Going forward, you can always reach for the general addition rule. If the events happen to be mutually exclusive, the overlap term is simply zero and the subtraction disappears on its own.

Side-by-side illustration contrasting overlapping events with a shared region versus mutually exclusive events with no shared region, showing that the simpler rule is a special case of the general addition rule

The general addition rule as presented covers two events, but real-world problems sometimes involve three or more. The idea extends naturally — you still add, then correct for every pairwise overlap — but each new event introduces additional overlaps to track. For three events $A$ , $B$ , and $C$ , the formula becomes:

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

Notice the pattern: you subtract every two-way overlap (because each was double-counted) and then add back the three-way overlap (because it was removed one too many times by the subtractions). This alternating add-subtract pattern is known as the inclusion–exclusion principle and generalizes to any number of events. In this course we will focus on the two-event version, but knowing that a systematic extension exists is valuable when you encounter more complex scenarios in practice.

As you start applying the general addition rule on your own, keep these pitfalls in mind:

Forgetting to subtract the overlap. This is the most frequent error and always produces an answer that is too high. If your result seems surprisingly large, double-check whether you subtracted $P(A \text{ and } B)$ .
Subtracting when there is no overlap. Sometimes a problem states that the events are mutually exclusive. In that case the overlap is zero — there is nothing to subtract. Inserting a nonzero value by mistake will push the answer too low.
Confusing "and" with "or." In probability, "or" means at least one of the two events occurs, while "and" means both occur simultaneously. Keeping these two words straight is essential for placing each number in the correct spot in the formula.

In this lesson, you turned the overlap insight from Lesson 3 into a working tool. The general addition rule — $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$ — corrects double-counting by subtracting the shared outcomes exactly once. You also confirmed that the simpler rule for mutually exclusive events is just a special case where the overlap is zero.

Up next, you will put the general addition rule to work in a series of hands-on practice tasks covering workplace surveys, marketing campaigns, gym memberships, and more. Dive in and build real confidence with the formula!