0
0

Complementary Events

Introduction

In probability, every event has a complementary event - an event that represents all outcomes in which the original event does not occur. Understanding complementary events is vital for solving problems where it’s easier to calculate what doesn’t happen rather than what does.

This pattern helps simplify probability calculations by using the fundamental rule: P(not E) = 1 - P(E).

Pattern: Complementary Events

Pattern

The probability of an event not occurring is equal to one minus the probability of the event occurring.

Formula: P(not E) = 1 - P(E)

Since probabilities always sum to 1, this relationship allows quick calculation of one when the other is known.

Step-by-Step Example

Question

A die is rolled once. Find the probability of not getting a 6.

Solution

  1. Step 1: List all possible outcomes

    Total possible outcomes on a die = {1, 2, 3, 4, 5, 6} → total = 6.

  2. Step 2: Identify favourable outcomes

    Favourable outcomes for getting a 6 = 1 (only the face ‘6’).

  3. Step 3: Compute probability of getting a 6

    P(getting a 6) = 1 / 6 = 1/6.

  4. Step 4: Apply complementary rule

    P(not getting a 6) = 1 - 1/6 = 5/6.

  5. Final Answer:

    5/6

  6. Quick Check:

    P(getting 6) + P(not getting 6) = 1/6 + 5/6 = 1 ✅

Quick Variations

1. “At least one” type problems (e.g., P(at least one head) = 1 - P(no heads)).

2. Finding the probability of not drawing a particular card or color.

3. Situations involving “none”, “not happening”, or “failure” outcomes.

Trick to Always Use

  • Step 1: Find the probability of the event occurring (P(E)).
  • Step 2: Subtract from 1 to get P(not E).
  • Step 3: Always verify that P(E) + P(not E) = 1.

Summary

Summary

In the Complementary Events pattern:

  • P(not E) = 1 - P(E).
  • Use this when it’s easier to find the opposite event.
  • Always check total probability = 1 for validation.
  • Common in “at least one” and “none” type problems.

Practice

(1/5)
1. A box contains 7 white balls and 3 black balls. One ball is drawn at random. What is the probability that the ball is not black?
easy
A. 7/10
B. 3/10
C. 1/2
D. 4/5

Solution

  1. Step 1: Identify total outcomes

    Total balls = 7 white + 3 black = 10.

  2. Step 2: Identify P(black)

    P(black) = 3 / 10 = 3/10.

  3. Step 3: Use complement

    P(not black) = 1 - P(black) = 1 - 3/10 = 7/10.

  4. Final Answer:

    7/10 → Option A.

  5. Quick Check:

    Not black = white count 7 → 7/10 ✅
Hint: Find P(E) for the unwanted event first, then subtract from 1.
Common Mistakes: Forgetting to include all balls when computing the total.
2. Two fair coins are tossed. What is the probability of getting at least one Head?
easy
A. 3/4
B. 1/4
C. 1/2
D. 1

Solution

  1. Step 1: Identify total outcomes

    Sample space for two coins = {HH, HT, TH, TT} → 4 outcomes.

  2. Step 2: Identify complement (no Heads)

    Complement = {TT} → P(no Heads) = 1/4.

  3. Step 3: Apply complement rule

    P(at least one Head) = 1 - P(no Heads) = 1 - 1/4 = 3/4.

  4. Final Answer:

    3/4 → Option A.

  5. Quick Check:

    Direct count: {HH, HT, TH} = 3 favourable out of 4 → 3/4 ✅
Hint: For 'at least one' problems, compute P(none) and subtract from 1.
Common Mistakes: Counting HT and TH as one outcome (they are distinct) leading to wrong totals.
3. One card is drawn from a standard 52-card deck. What is the probability that the card is not a Spade?
easy
A. 1/4
B. 3/4
C. 1/2
D. 13/52

Solution

  1. Step 1: Identify total outcomes

    Total cards = 52.

  2. Step 2: Identify P(Spade)

    Number of spades = 13 → P(Spade) = 13/52 = 1/4.

  3. Step 3: Use complement

    P(not Spade) = 1 - P(Spade) = 1 - 1/4 = 3/4.

  4. Final Answer:

    3/4 → Option B.

  5. Quick Check:

    Spades = 13, not spade = 39 → 39/52 = 3/4 ✅
Hint: P(not E) = 1 - P(E); for suits P(E) often equals 13/52 = 1/4.
Common Mistakes: Using 26 (red cards) values instead of correctly identifying the suit count.
4. A bag contains 4 red and 6 blue balls. One ball is drawn at random. What is the probability that the drawn ball is not blue?
medium
A. 3/5
B. 7/10
C. 2/5
D. 4/6

Solution

  1. Step 1: Identify total outcomes

    Total balls = 4 red + 6 blue = 10.

  2. Step 2: Identify P(blue)

    P(blue) = 6 / 10 = 3/5.

  3. Step 3: Use complement

    P(not blue) = 1 - P(blue) = 1 - 3/5 = 2/5.

  4. Final Answer:

    2/5 → Option C.

  5. Quick Check:

    Not blue = red count 4 → 4/10 = 2/5 ✅
Hint: Either compute 1 - P(blue) or directly use favourable count for 'not' event.
Common Mistakes: Forgetting to compute total before applying complement.
5. Three fair coins are tossed. What is the probability that the outcome is not exactly one Head?
medium
A. 3/8
B. 1/2
C. 5/16
D. 5/8

Solution

  1. Step 1: Identify total outcomes

    Total outcomes for 3 coins = 2³ = 8.

  2. Step 2: Identify P(exactly one Head)

    Exactly one Head outcomes = {HTT, THT, TTH} → 3 outcomes → P(exactly one Head) = 3/8.

  3. Step 3: Use complement

    P(not exactly one Head) = 1 - 3/8 = 5/8.

  4. Final Answer:

    5/8 → Option D.

  5. Quick Check:

    Other outcomes count = 8 - 3 = 5 → 5/8 ✅
Hint: Compute the unwanted case count (exactly one) and subtract from 1.
Common Mistakes: Miscounting arrangements for exactly one Head (forgetting permutations).

Mock Test

Ready for a challenge?

Take a 10-minute AI-powered test with 10 questions (Easy-Medium-Hard mix) and get instant SWOT analysis of your performance!

10 Questions
5 Minutes