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Complementary Events are a type of event in which one outcome can only happen when the other does not. Each of these events complements the other.
- Complementary Events are a measurement of the occurrence of an event.
- It is an important topic included in the topic probability.
- The combination of event and complementary event help in defining the Bernoulli Trial.
- Complementary Events are mutually exclusive and collectively exhaustive.
- It is denoted by A′, Ac, or A.
- Mathematically, it can be represented as:
P(A) + P(A') = 1
- The concept can be used in our day-to-day lives.
- Students preparing for the examination can clear the exam or not clear the exam.
- It may or may not rain.
Key Terms: Complementary Events, Probability, Statistics, Outcomes, Sample Space, Events, Complement, Mutually Exclusive, Collectively Exhaustive, Set, Experimental Probability
What are Complementary Events in Probability?
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Complementary Events as name suggest is a type of event when probability is based on the occurrence of two outcomes. Complements are outcomes or outcomes in which an event does not take place.
- The probability of complementary events adds up to 1 in an intuitive way.
- Two complimentary events can’t be occurring at the same time.
- If their probabilities add up to one, they are considered specific events as well.
- Suppose there is an event named A.
- The complement of the required event is represented as A'.
- A' is sometimes written without an apostrophe.
- They are all equivalent and express the same relationship in different ways.
- They primarily refer to each other as the complement rule.
- For any event A, A' represents the remaining values of the sample space S.
- It can be represented as:
A’ = S – A
Example of What are Complementary Events in Probability?Example: Let's take a closer look at the events highlighted above. A person who does not go to school is a complement to someone who does.
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Complimentary Events
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Complementary Events Properties
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For any two events to be complementary, it must satisfy following two properties:
Mutually Exclusive
Two events must be mutually exclusive to each other to form a complement event. It means they are not occurring at the same time and are disjoint to each other.
Collectively Exhaustive
Two events must be collectively exhaustive to each other to form a complementary event. It means that both the event and its complement must form the sample space.
Rules of Complementary Events
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The rules of complementary events states that sum of probability of an event and probability of the complementary event is equivalent to one.
- Suppose there is an event named A.
- The complement of the required event is represented as A'.
- P(A) represent the probability of an event.
- P(A’) represent the probability of the complementary event.
- It can be represented as:
P(A) + P(A') = 1
P(A) = 1 - P(A')
P(A') = 1 - P(A)
Example of Rules of Complementary EventsExample: There are 10 balls in a bag out of which 4 are black, 2 are red, 2 is blue, 1 are pink, and 1 are purple. Let X be the event of selecting a primary color. Find P(X'). |
Complementary Events Formulas
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By using the complement rule, we can estimate the probability of complementary events. It is given as:
P(A) + P(A') = 1
Where:
- 'A' is the complement of an event, while 'A' is the event itself
Example of Complementary Events FormulasExample: Let's say the probability of an individual going to school is 7/8. In that case, the likelihood that they do not go is 1–7/8=1/8. It can be expressed as follows. Let A be the occasion where you go to school, and then: P(A)=7/8 P(A’)=1–7/8=1/8 |
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| Class 10 Maths Related Concepts | ||
|---|---|---|
| Relation and Function | Analysis of Variance | Mode of Grouped Data |
| Probability — A Theoretical Approach | Types of events | Remainder Theorem |
Benefits of Complementary Events
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The benefits if complementary events are as follows:
- Complementary Events is helpful when the sample space includes a significant number of outcomes.
- It is beneficial to be familiar with this rule.
- For the sake of clarity, let S be the event of getting two unique digits when throwing two dice.
- Then, when finding P(S), it is easier to find the probability of S', the event that the result doubles, then subtract it from 1.
| Example: The values of S' are (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6). P(S’)=6/6×6=6/36=1/6 Therefore: P(S)=1–1/6=5/6 |
We only had to determine the following outcomes in the sample space of S and of the experiment:
- The number of consequences for S'
- Approximately how many outcomes there are in the experimental sample space
Depending on the type of probability desired, it can be helpful to list the outcomes of the event or its complement.
Things to Remember
- Complementary event depend upon the type of probability desired.
- It can be helpful to list the outcomes of the event or its complement
- A given event occurs with a probability of 1.
- A probability of an impossible event is zero.
- There is a probability E of happening, i.e., 0≤P(E)≤1.
- An elementary event has only one outcome.
- During an experiment, the probability of all the elementary events is 1.
Sample Questions
Ques: What is the probability of not getting a 3 if you roll a dice? (3 marks)
Ans: There are 6 events that can occur. We can get 1,2,3,4,5,6 when we roll a dice.
- P(getting 3) = 1/6
- Since both the events are complimentary.
- Therefore P(getting 3) + P(not getting 3) = 1
- P(not getting 3) = 1-1/6
- P(not getting 3) = 5/6
Ques: In probability, what is the formula for complementary events? (2 marks)
Ans: To apply the rule of complementary events, you need to add the probability of something happening, and it was not happening. That equals 100% (or 1 in decimal form). In other words, if there is a 40% chance of rain, there must be a 60% chance of no rain. Therefore, 40% + 60% = 100%.
Ques: A bag contains both red and blue balls. If the probability of getting a red ball is 2/5. What is the probability of getting blue balls? (2 marks)
Ans: Since we know that both are complimentary events. Therefore P(red balls) + P(blue balls) = 1
So, P(blue balls) = 1-P(red balls)
= 1 - 2/5
=3/5
Ques: How likely are simple events, compound events, and complimentary events? (2 marks)
Ans: Calculating the probability of a single event is called the probability of simple events. Using the formula: the number of favorable outcomes over the number of total products, we can find the probability of an event occurring. Complex events involve the possibility of multiple events taking place simultaneously.
Ques: What is the probability of not getting a white ball, if the probability of getting a white ball is 1/4? (2 marks)
Ans: Given to us that,
- P(white balls)= 1/4
- Therefore, P(not white balls) = 1- P(white balls)
- 1-1/4
- ¾
Ques: What is the probability of not getting a 5 if you roll a dice? (3 marks)
Ans: There are 6 events that can occur. We can get 1,2,3,4,5,6 when we roll a dice.
- P(getting 5) = 1/6
- Since both the events are complimentary.
- Therefore P(getting 5) + P(not getting 5) = 1
- P(not getting 5) = 1-1/6
- P(not getting 5) = 5/6
Ques: There are 20 balls in a bag out of which 13 are black, 2 are red, 1 is blue, 2 are pink, and 2 are purple. Let X be the event of selecting a primary color. Find P(X')? (3 marks)
Ans: X = {13 black, 1 blue}
- Total balls = 20
- Number of favorable outcomes = 14
- P(X) = 14/ 20
- Using the rule of complementary events, P(A') = 1 - P(A)
- P(X') = 1 - (14 / 20) = 6 / 20
Ques: A random number is chosen from 1 to 30. Calculate the probability of not choosing a perfect square? (2 marks)
Ans: Let Z' be the event of choosing a perfect square. The sample space is given as follows:
- Z' = {1, 4, 9, 16, 25}
- Total number of outcomes = 30
- Favorable outcomes = 5
- P(Z') = 5 / 30.
- P(Z) = 1 - (5 / 30)
- 1- 1/6
- 5/6
Ques: What is the probability of not getting a red ball, if the probability of getting a red ball is 1/10? (2 marks)
Ans: Given to us that,
- P(white balls)= 1/10
- Therefore, P(not white balls) = 1- P(white balls)
- 1-1/10
- 9/10
Ques: What is the probability of not getting a 2 if you roll a dice? (3 marks)
Ans: There are 6 events that can occur. We can get 1,2,3,4,5,6 when we roll a dice.
- P(getting 2) = 1/6
- Since both the events are complimentary.
- Therefore P(getting 2) + P(not getting 2) = 1
- P(not getting 2) = 1-1/6
- P(not getting 2) = 5/6
Ques: There are 20 balls in a bag out of which 3 are black, 12 are red, 1 is blue, 3 are pink, and 1 are purple. Let X be the event of selecting a primary color. Find P(X')? (3 marks)
Ans: X = {3 black, 1 pink}
- Total balls = 20
- Number of favorable outcomes = 4
- P(X) = 4/ 20
- Using the rule of complementary events, P(A') = 1 - P(A)
- P(X') = 1 - (4 / 20) = 16 / 20
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