Complementary Events: Formula, Rules and Examples

Complementary Events are pairs of events in which one event occurs exactly when the other does not. Their probabilities add up to 1 because the two events are mutually exclusive and together cover the entire sample space. This page explains the complementary events formula, properties, solved examples and methods used in probability questions.

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What Are Complementary Events?

Two events are complementary when one event occurs if and only if the other event does not occur. If event A has complement A′, then A and A′ are mutually exclusive and exhaustive.

The complement of A is written as A′, Aᶜ or \bar{A}. The events have no common outcome, so P(A ∩ A′) = 0, and every outcome belongs to either A or A′, so P(A ∪ A′) = 1. Therefore, P(A) + P(A′) = 1. For example, when a die is rolled, getting an even number and not getting an even number are complementary events.

Complementary Events Video Lessons

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Complementary and Mutually Exclusive Events

Understand complementary and mutually exclusive events in probability, including their definitions, differences, and how they relate to possible outcomes in an experiment.

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Complementary Events Formula & Tricks

Important Formulas

Complementary events formula
P(A′) = 1 − P(A)

The probability of an event's complement equals 1 minus the probability of the event.

Sum of complementary probabilities
P(A) + P(A′) = 1

A complementary pair covers all possible outcomes and has no overlap.

Complement of a union
P((A ∪ B)′) = 1 − P(A ∪ B)

This gives the probability that neither A nor B occurs.

At least one occurrence
P(at least one) = 1 − P(none)

Use the complement when the probability of no occurrence is easier to calculate.

Quick Tricks

Use the complement for ‘at least one’

For repeated independent trials, calculate the probability of zero successes first, then subtract it from 1.

Example: The probability of getting at least one six in two die rolls is 1 − P(no six) = 1 − (5/6)² = 11/36.
Check both complement conditions

A pair is complementary only when the events cannot occur together and one of them must occur. Merely being different events is not enough.

Example: On a die, ‘odd’ and ‘even’ are complementary. ‘Getting 1’ and ‘getting 2’ are mutually exclusive but not complementary because outcomes 3, 4, 5 and 6 remain.

Complementary Events Concepts

Identifying a Complementary Pair

Events A and A′ are complementary if they are mutually exclusive and collectively exhaustive.

Mutually exclusive means A ∩ A′ = ∅, so both events cannot occur in the same trial. Collectively exhaustive means A ∪ A′ = S, where S is the sample space. The complement is commonly formed by adding words such as ‘not’, ‘does not’ or ‘fails to’.

Example: For a card drawn from a standard deck, ‘drawing a heart’ and ‘not drawing a heart’ form a complementary pair.

Finding the Probability of a Complement

Subtract the given event probability from 1 to find the probability of its complement.

If P(A) = p, then P(A′) = 1 − p. For a fraction, retain the same denominator when convenient; for a percentage, subtract the percentage from 100%.

Example: If P(A) = 3/8, then P(A′) = 1 − 3/8 = 5/8. If P(A) = 42%, then P(A′) = 58%.

Complement of ‘At Least’ and ‘At Most’ Events

The complement of ‘at least one’ is ‘none’, while the complement of ‘at most k’ is ‘more than k’.

For any event count, the complement reverses the inequality appropriately: the complement of X ≥ k is X < k, and the complement of X ≤ k is X > k. Thus, P(X ≥ 1) = 1 − P(X = 0).

Example: The complement of getting at most 2 heads is getting at least 3 heads, when the number of trials is fixed.

Complement in Repeated Independent Trials

For n independent trials, the probability of no occurrence of an event with probability p in one trial is (1 − p)ⁿ.

Therefore, the probability of at least one occurrence is 1 − (1 − p)ⁿ. Independence is required because the probability of the combined outcomes is found by multiplication.

Example: If a machine produces a defective item with probability 0.1 on each independent trial, the probability of at least one defective item in three trials is 1 − (0.9)³ = 0.271.

Complementary Events Visual Guide & Data

Use the table and chart below to understand the data better.

Complementary Event Properties

Basic probability properties of an event and its complement

Property Expression Meaning
No common outcome A ∩ A′ = ∅ A and A′ cannot occur together
All outcomes covered A ∪ A′ = S One of A or A′ must occur
Probability sum P(A) + P(A′) = 1 Their probabilities total 1
Complement of complement (A′)′ = A Taking the complement twice returns A

Note: S denotes the sample space and ∅ denotes the empty set.

Event A and Its Complement

Visual guide: Event A and Its Complement
A and A′ are disjoint and together form the complete sample space

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Practice Complementary Events Questions

Practise published questions related to this topic.

1Pure ghee costs ₹120 per kg. After adulterating it with vegetable oil costing ₹60 per kg, a shopkeeper sells the mixture at the rate of ₹98 per kg, thereby making a profit of 40%. In what ratio does he mix the two?→ 2When reduced by 6, the smallest number M is divisible by 9, 12, and 26. The digit sum of M is:→ 3If on a marked price, the difference of selling price with a discount of 35% and two successive discounts of 20% and 15% is ₹90, find the marked price (in ₹).→ 4A dishonest shopkeeper pretends to sell his goods at cost price but uses wrong weights (tolas) and makes a profit of 14 2/7 % . For a weight of 1 kg he uses ________ weights.→ 5Shubh marks an article 40% above its cost price. He allows a single discount of d% on the marked price and still makes a profit of 10% on the cost price. Find the discount d%.→ 651/4 + 551/4 + 5551/4 = ?→ 7A man bought 100 items for ₹2,500. Out of these, 20 items were damaged and could not be sold. He sold the remaining items at ₹30 each. Find the loss percentage.→ 8A, B, and C start a business by investing ₹60,000, ₹80,000, and ₹1,20,000, respectively. After one year, the total profit earned is ₹78,000. By how much does C's share exceed B's share?→ 9Simplify: 30 + [{-9 × (26 - 11 + 3)}]→ 10The sum of two numbers is 54 and their LCM is 195. The two numbers are:→

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Complementary Events Quick Quiz

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Quick Revision Notes

Complementary Events Revision Points

Use these rules to revise complementary events and their probability calculations.

  • P(A′) = 1 − P(A).
  • P(A) + P(A′) = 1.
  • A and A′ are mutually exclusive: A ∩ A′ = ∅.
  • A and A′ are exhaustive: A ∪ A′ = S.
  • The complement of ‘at least one’ is ‘none’.
  • For n independent trials, P(at least one occurrence) = 1 − (1 − p)ⁿ.
  • The complement of A′ is A.

Complementary Events FAQs

What is the complementary events formula?

The formula is P(A′) = 1 − P(A), or equivalently P(A) + P(A′) = 1.

If the probability of rain is 0.35, what is the probability of no rain?

No rain is the complement of rain, so P(no rain) = 1 − 0.35 = 0.65.

Are mutually exclusive events always complementary?

No. Mutually exclusive events cannot occur together, but complementary events must also cover every outcome in the sample space. ‘Getting 1’ and ‘getting 2’ on a die are mutually exclusive but not complementary.

What is the complement of getting at least one head in three coin tosses?

Its complement is getting no heads, which means getting three tails.

What is the probability of at least one success in four independent trials if the success probability is 1/5?

P(at least one) = 1 − P(no success) = 1 − (4/5)⁴ = 369/625.

What is the complement of an event with probability 72%?

The complement has probability 100% − 72% = 28%.

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