Understanding how to calculate the complementary probability is crucial in the field of probability and statistics. This concept helps us quantify the likelihood of an event not occurring, which is just as important as calculating the chance that it does happen.

Introduction to Complementary Probability

Complementary probability involves understanding the likelihood of an event not occurring. It's a key concept in probability theory, essential for various real-world applications and problem-solving in exams.

What is Complementary Probability?

Complementary probability is calculated by subtracting the probability of an event from 1. This approach is based on the principle that the total probability of all possible outcomes in a given situation is always 1 (or 100%).

• Formula: $P(A') = 1 - P(A)$
  • Where $P(A)$ is the probability of the event happening, and $P(A')$is the probability of the event not happening.

Image courtesy of Cuemath

Importance of Complementary Probability

• Simplifies calculations: Especially useful when calculating the probability of an event not happening is more straightforward than calculating the probability of it happening.
• Essential in probability distributions: Helps in understanding and applying various probability distributions.

Worked Examples

Example 1: Probability of Not Drawing an Ace from a Deck of Cards

Consider a standard deck of 52 cards. The probability of drawing an ace $P(\text{Ace})$ is 4 out of 52, since there are 4 aces in the deck.

• Given: $P(Ace) = \dfrac{4}{52}$
• Find: $P(Not\ Ace) = ?$

Solution:

1. Calculate $P(Ace)$: $P(Ace) = \dfrac{4}{52}$

2. Apply complementary probability formula: $P(Not\ Ace) = 1 - P(Ace)$

3. Substitute the value of $P(Ace)$: $P(Not\ Ace) = 1 - \dfrac{4}{52} = \dfrac{48}{52}$

4. Simplify: $P(Not\ Ace) = \dfrac{12}{13}$

Example 2: Probability of Not Getting a Head in a Coin Toss

When tossing a fair coin, the probability of getting a head $P(Head)$ is 1 out of 2.

• Given: $P(Head) = \dfrac{1}{2}$
• Find: $P(Not\ Head) = ?$

Solution:

1. Calculate $P(Head)$: $P(Head) = \dfrac{1}{2}$

2. Apply complementary probability formula: $P(Not\ Head) = 1 - P(Head)$

3. Substitute the value of $P(Head)$: $P(Not\ Head) = 1 - \dfrac{1}{2} = \dfrac{1}{2}$

Practice Problems

To solidify your understanding, solve these practice problems using the complementary probability formula.

Problem 1

In a bag of 30 balls, 9 are red. What is the probability of picking a ball that is not red?

Solution:

• Given: Total balls = 30, Red balls = 9
• Find: Probability of not picking a red ball

Steps:

1. Calculate $P(Red)$: $P(Red) = \dfrac{9}{30}$

2. Apply complementary probability formula: $P(Not\ Red) = 1 - P(Red)$

3. Substitute the value of $P(Red)$: $P(Not\ Red) = 1 - \dfrac{9}{30} = \dfrac{21}{30}$

4. Simplify: $P(Not\ Red) = \dfrac{7}{10}$

Problem 2:

A dice is rolled once. What is the probability that it does not land on 6?

Solution:

• Given: $P(6) = \dfrac{1}{6}$ (Since a die has 6 faces and only one of them is a 6)
• Find: $P(Not\ 6) = ?$

Steps:

1. Apply complementary probability formula: $P(Not\ 6) = 1 - P(6)$

2. Substitute the value of $P(6)$: $P(Not\ 6) = 1 - \dfrac{1}{6} = \dfrac{5}{6}$