Addition Theorem of Probability and Events | NIMCET

Quick answer: The addition theorem states $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ . For mutually exclusive events, $P(A \cap B) = 0$ , so $P(A \cup B) = P(A) + P(B)$ . For three events, $P(A \cup B \cup C) = \sum P(A) - \sum P(A \cap B) + P(A \cap B \cap C)$ . The probability that exactly one of A, B occurs is $P(A) + P(B) - 2P(A \cap B)$ .

This is the probability version of the inclusion-exclusion principle, building on probability basics.

Two events

Quantity	Formula
A or B	$P(A) + P(B) - P(A \cap B)$
Only A	$P(A) - P(A \cap B) = P(A \cap B')$
Exactly one	$P(A) + P(B) - 2P(A \cap B)$
Neither	$1 - P(A \cup B) = P(A' \cap B')$
Not both	$1 - P(A \cap B)$

Three events

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

Mutually exclusive and exhaustive events

Type	Condition	Consequence
Mutually exclusive	$A \cap B = \phi$	$P(A \cup B) = P(A) + P(B)$
Exhaustive	$A \cup B = S$	$P(A \cup B) = 1$
Both	Partition of S	$P(A) + P(B) = 1$

For pairwise mutually exclusive and exhaustive events $E_1, \ldots, E_n$ :

$P(E_1) + P(E_2) + \cdots + P(E_n) = 1$

Useful bounds

Bound	Statement
Union	$\max(P(A), P(B)) \le P(A \cup B) \le P(A) + P(B)$
Intersection	$P(A) + P(B) - 1 \le P(A \cap B) \le \min(P(A), P(B))$

These bounds show whether given values are consistent.

Worked examples

Example 1: a card

Find the probability that a card drawn from a pack is a king or a heart.

$\frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}$

Example 2: using the formula backwards

If $P(A) = 0.5$ , $P(B) = 0.4$ and $P(A \cup B) = 0.7$ , find $P(A \cap B)$ and $P(\text{exactly one})$ .

Step 1: $P(A \cap B) = 0.5 + 0.4 - 0.7 = 0.2$ .

Step 2: Exactly one $= 0.5 + 0.4 - 0.4 = 0.5$ .

Example 3: divisibility

A number is chosen from 1 to 100. Find the probability that it is divisible by 2 or 5.

$\frac{50}{100} + \frac{20}{100} - \frac{10}{100} = \frac{60}{100} = \frac{3}{5}$

Example 4: mutually exclusive and exhaustive

Three mutually exclusive and exhaustive events have $P(A) = 2P(B) = 3P(C)$ . Find $P(A)$ .

Step 1: Let $P(A) = p$ . Then $P(B) = \frac{p}{2}$ and $P(C) = \frac{p}{3}$ .

Step 2: $p\left(1 + \frac{1}{2} + \frac{1}{3}\right) = 1$ , so $p = \frac{6}{11}$ .

Example 5: checking consistency

Can $P(A) = 0.6$ , $P(B) = 0.7$ and $P(A \cap B) = 0.2$ ?

The minimum possible $P(A \cap B)$ is $0.6 + 0.7 - 1 = 0.3$ , so no.

Summary table

Idea	Key result
Addition theorem	$P(A) + P(B) - P(A \cap B)$
Mutually exclusive	Just add
Exactly one	$P(A) + P(B) - 2P(A \cap B)$
Neither	$1 - P(A \cup B)$
Partition	Probabilities sum to 1

Exam tips

Translate words into set notation first: "or" is ∪, "and" is ∩, "not" is complement.
Check consistency questions with the intersection bounds.
For "divisible by a or b", use the count of multiples of a, of b, and of their LCM.

Next, study conditional probability and independent events, or see the NIMCET Probability and Statistics chapter.

Common mistakes

Adding probabilities of events that are not mutually exclusive.
Confusing mutually exclusive with independent.
Using the product of numbers instead of their LCM for divisibility overlaps.
Forgetting to add back the triple intersection for three events.

Frequently Asked Questions

What is the addition theorem of probability?

P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

What is P(A ∪ B) for mutually exclusive events?

P(A) + P(B), because P(A ∩ B) = 0.

What is the probability that exactly one of A and B occurs?

P(A) + P(B) − 2P(A ∩ B).

What are exhaustive events?

Events whose union is the whole sample space, so at least one of them must occur.

Can events be both mutually exclusive and exhaustive?

Yes. For example, getting an even or an odd number on a die. Their probabilities then add up to 1.