Complimentary Events

The probability of an event is defined as the measure of the chance of occurrence of an event. The set of outcomes from a random experiment is called an event. Complementary event is a type of event in probability. You can refer to the following sections to get the complete details of the complimentary events in probability. Check the examples of complementary events and solved questions for a better understanding of the concept.

What is a Complimentary Event?

For any event E, there exists another event E’ which contains the remaining elements of the sample space A. E’ is the complementary event of E. It can also be defined as the non-occurrence of an event. Mathematically, it is represented as:

E’ = S – E

Events E, E’ are mutually exclusive and exhaustive.

The complement of an event is represented as E’ or E or Ec.

If P(E) is the probability of an event and P(E’) is the probability of the complementary event E’, then P(E) + P(E’) = 1. So, the probabilities of two complimentary events add up to 1.

Examples of Complementary Events

The following listed are examples of the complement of an event.

  • When a coin is tossed, the probability of getting ‘tail’ and getting ‘head’ are a complement to each other.
  • When two coins are flipped simultaneously, getting ‘at least one head’ and ‘no head’ are complementary events.
  • When a die is thrown:
    • Obtaining ‘even face’ and ‘odd face’ are complementary events of each other.
    • Getting ‘multiple of 2’ and ‘not multiple of 2’ are complement events.
    • Getting ‘divisible by 3’ is a complement event of getting ‘not divisible by 3’ event.

Questions on Complement of an Event

Question 1:
The probability of getting a red ball from a bag of balls is \(\frac { 3 }{ 4 } \). What is the probability of not getting a red ball?

Solution:
Given that,
P(red ball) = \(\frac { 3 }{ 4 } \)
P(not red ball) = 1 – \(\frac { 3 }{ 4 } \)
= \(\frac { 1 }{ 4 } \)
Therefore, the probability of not getting a red ball from a bag of balls is \(\frac { 1 }{ 4 } \).

Question 2:
A box contains white and black marbles. The probability of getting a white marble from the box of marbles is \(\frac { 7 }{ 10 } \). What is the probability of getting a black marble?

Solution:
Let A be the event of getting a white marble and B be the event of getting a black marble
P(B) is the probability of getting a black marble which is equal to the probability of not getting a white marble
So, probability of not getting white marble = P(B) = 1 – P(A)
= 1 – \(\frac { 7 }{ 10 } \)
= \(\frac { 3 }{ 10 } \)
Therefore, the probability of getting a black marble is \(\frac { 3 }{ 10 } \).

Question 3:
In a laptop shop, there are 32 defective laptops out of 250 laptops. If one laptop is taken out at random from this laptop shop, what is the probability that it is a non-defective laptop?

Solution:
The total number of laptops in laptop shop = 250
The number of defective laptops = 32
Let A be the event of getting a defective laptops and
E be the event of getting a non defective laptops
P(A) = probability of getting a defective laptop
= \(\frac { 32 }{ 250 } \) = 0.128
Therefore, the probability of getting a non defective laptop = 1 – P(A) = 1 – 0.128
= 0.872

FAQ’s on Complementary Events in Probability

1. What are the complements of events?

Events will be complements when there are two outcomes like getting a job or not getting a job. The complement of an event is the exact opposite of the event.

2. What is the difference between mutually exclusive and complementary events?

Complementary events will become mutually exclusive events when their combination is the sample space.

3. What is the addition of two events that are complementary?

If two events are complementary, then the sum of their probabilities is 1.

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