Content Writer
Probability is an important concept of mathematics which deals with how likely it is for an event to occur.
- In daily conversations, we often use probabilistic statements such as “It might rain today”, “I will most probably pass the exam as the exam was not too tough”, or “Most likely, he will be selected”.
- An event in probability is a collection of an experiment's results.
- In simple terms, the event is the subset of the corresponding sample space.
- Sample space is defined as a collection of probable outcomes of an experiment.
- Capital letters are used to denote the algebra of events.
- The throwing of a die, tossing of a coin, or lottery draws are all instances of random events.
- Any subset, say E, of a sample space S is referred to as an event in probability.
- The article will help you learn about the events in probability, the types of events, how they are classified and how the algebra of events works.
| Table of Content |
Key Terms: Probability, Algebra of Events, Complementary Event, Event ‘A or B’, Event ‘A and B’, Event ‘A but not B’, Sample Space, Events
Algebra of Events in Probability
[Click Here for Sample Questions]
An event refers to a set of outcomes that occur as a result of an experiment. There are different ways to calculate probability based on the nature of the event and the available information. The algebra of events in probability includes
- Complementary Event
- The Event ‘A or B’
- The Event ‘A and B’
- The Event ‘A but not B’
Read Also:
| Mathematics Related Topics | ||
|---|---|---|
| Onto Function | Scalar matrix | Signum Function |
| Identity matrix | Rolle’s Theorem | Derivative of Inverse Trignometric Functions |
Complementary Event
[Click Here for Sample Questions]
Every event A has a corresponding event A′, which is also known as the complementary event of A or the event 'not A'.
A′ = {ω : ω ∈ S and ω ∉A} = S – A
- In this equation, A represents the event
- ω represents the outcome associated with the event A, and S is the sample space.
- The complimentary event can be represented by a Venn diagram as follows:
Complementary Events
Example of Complementary EventsExample: If we consider the experiment of tossing three coins, the sample space can be represented as: S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
|
The Event ‘A or B’
[Click Here for Sample Questions]
The event "A or B" is the union of two sets, A and B. It is denoted by A ∪ B and contains all the elements that are in set A or B or both.
- If sets A and B are two events associated with a sample space, then A ∪ B is either A or B or both.
- This event, "A or B", is also known as "A union B".
- Therefore, the event "A or B" can be written as
A ∪ B = {ω: ω ∈ A or ω ∈ B}
- The Venn diagram representation of A or B, i.e. A ∪ B, is given below:
The Event ‘A or B’
Example of The Event ‘A or B’Example: Let's consider the experiment of throwing two dice at a time. The associated sample space can be written as: S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
|
The Event ‘A and B’
[Click Here for Sample Questions]
The intersection of two sets, A and B, is denoted as A ∩ B. The Event ‘A and B’ refers to the set of elements that belong to both A and B.
- In other words, it is the set of common elements to both A and B.
- Therefore, the event "A and B" can be written as
A ∩ B = {ω: ω ∈ A and ω ∈ B}
- This is shown in the Venn diagram below.
The Event ‘A and B’
Example of The Event ‘A and B’Example: Consider the experiment of throwing two dice at a time. The sample space associated with this experiment can be represented as S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}.
|
The Event ‘A but not B’
[Click Here for Sample Questions]
The set A – B, or difference of sets A and B, is the occurrence of A but not B. This means it consists of the elements that are in A but not in B.
- The event A but not B can be calculated using the following formula
A - B = A ∩ B', which means that A - B = {ω: ω ∈ A and ω ∉ B}
- The event A but not B is shown in the Venn diagram below.
The Event ‘A but not B’
Example of The Event ‘A but not B’Example: Consider a sample space S = {1, 2, 3, 4, 5, 6}. Let A be the set of prime numbers, and B be the set of an odd number.
|
Things to Remember
- The results or outcomes of a random experiment are called events.
- For example, when an unbiased coin is tossed, the events are "head" and "tail".
- Algebra of events are denoted by capital letters like A, B, C, and so on.
- Theoretical probability is the likelihood of an event occurring based on mathematical rules or assumptions.
- Subjective probability is the relative probability of an event based on the beliefs, opinions, or biases of an individual about the likelihood of the event occurring.
- Empirical probability is calculated based on observations and experiments.
Also Read:
Sample Questions
Ques: Suppose there are 4 red, 7 blue and 9 green balls in a bag. Two balls are drawn at random from the bag. Find the probability that the two balls drawn are red? (2 marks)
Ans: Total number of balls in the bag = 4 + 7 + 9 = 20.
- Two balls are drawn at random.
- The total number of ways in which any two balls are drawn = 20C2 = 66.
- The number of favorable cases of getting two red balls = 4C2 = 6.
- Therefore, the required probability = \(\frac{6}{66}\) = \(\frac{1}{11}\).
Ques: X, Y, and Z are three mutually exclusive and exhaustive events of a random experiment. If P(X) = \(\frac{4}{5}\) P(Y) and P(Z) = \(\frac{3}{5}\) P(C). Find P(C)? (2 marks)
Ans: Since X, Y and Z are mutually exclusive and exhaustive events, P(X) + P(Y) + P(Z) = 1.
- \(\frac{4}{5}\) P(C) + \(\frac{3}{5}\) P(C) + P(C) = 1
- \(\frac{12}{5}\) P(C) = 1
- P(C) = \(\frac{5}{12}\).
Ques: Consider the experiment of throwing a die and the events are:
A: ‘a number less than 4 appears’,
B: ‘a number greater than 2 but less than 5 appears’
Write the sets representing the events (A) A or B (B) A and B (C) A but not B (D) ‘not A’ (E) ‘not B’? (5 marks)
Ans: Sample space = S = {1, 2, 3, 4, 5, 6}
- A = {1, 2, 3}
- B = {3, 4}
(A) A or B = A ⋃ B = {1, 2, 3} ⋃ {3, 4} = {1, 2, 3, 4}
(B) A and B = A ∩ B = {1, 2, 3} ∩ {3, 4} = {3}
(C) A but not B = A – B = {1, 2, 3} – {3, 4} = {1, 2}
(D) not A = A′ = S – A = {1, 2, 3, 4, 5, 6} – {1, 2, 3} = {4, 5, 6}
(E) not B = B′ = S – B = {1, 2, 3, 4, 5, 6} – {3, 4} = {1, 2, 5, 6}
Ques: In the game of Ludo, if a die is thrown, the event of getting even numbers is denoted by E1, while the event of getting a number more than 3 is represented by E2. Find the set of the following events: E1 or E2, E1 and E2? (2 marks)
Ans: S= {1, 2, 3, 4, 5, 6}
- E1 (only even numbers) = {2, 4, 6}
- E2 (number more than 3) = {4, 5, 6}
- E1 or E2 = {2, 4, 5, 6}
- E1 and E2 = {4, 6}
Ques: Write the sample space for tossing three coins at once, also answer the event of 2 exactly 2 heads at a time? (2 marks)
Ans: For tossing three coins the sample space are as follows:
- S = {(H, H, H), (H, H, T), (H, T, H), (T, H, H), (T, T, H), (T, H, T), (H, T, T), (T, T, T)}
- Hence, the sample space comprises 6 possible outcomes
- Event (E) for the occurrence of exactly two heads,
- E = {(H, H, T), (H, T, H), (T, H, H)}
Ques: Suppose there are 5 red, 7 blue and 2 green balls in a bag. Two balls are drawn at random from the bag. Find the probability that the two balls drawn are green? (2 marks)
Ans: Total number of balls in the bag = 5 + 7 + 2 = 14.
- Two balls are drawn at random.
- The total number of ways in which any two balls are drawn = 14C2 = 91.
- The number of favorable cases of getting two red balls = 2C2 = 1.
- Therefore, the required probability = \(\frac{1}{91}\)
Ques: A, V, and M are three mutually exclusive and exhaustive events of a random experiment. If P(A) = \(\frac{1}{6}\) P(V) and P(M) = \(\frac{5}{6}\) P(C). Find P(C)? (2 marks)
Ans: Since A, V and M are mutually exclusive and exhaustive events, P(A) + P(V) + P(M) = 1.
- \(\frac{1}{6}\) P(C) + \(\frac{5}{6}\) P(C) + P(C) = 1
- \(\frac{12}{6}\) P(C) = 1
- 2P(C) = 1
- P(C) = \(\frac{1}{2}\)
Ques: Consider the experiment of throwing a die and the events are:
A: ‘a number less than 7 appears’,
B: ‘a number greater than 3 but less than 5 appears’
Write the sets representing the events (i) A or B (ii) A and B (iii) A but not B (iv) ‘not A’ (v) ‘not B’? (5 marks)
Ans: Sample space = S = {1, 2, 3, 4, 5, 6, 7, 8}
A = {1, 2, 3, 4, 5, 6}
B = {4}
(i) A or B = A ⋃ B = {1, 2, 3, 4, 5, 6} ⋃ {4} = {1, 2, 3, 4, 5, 6}
(ii) A and B = A ∩ B = {1, 2, 3, 4, 5, 6} ∩ {4} = {4}
(iii) A but not B = A – B = {1, 2, 3, 4, 5, 6} – {4} = {1, 2, 3, 5, 6}
(iv) not A = A′ = S – A = {1, 2, 3, 4, 5, 6, 7, 8} – {1, 2, 3, 4, 5, 6} = { 7, 8}
(v) not B = B′ = S – B = {1, 2, 3, 4, 5, 6, 7, 8} – {4}= {1, 2, 3, 5, 6, 7, 8}
Ques: Given the sample space S = {10, 11, 12, 13, 14, 15, 16, 17}, and event E as all even numbers. What is the complementary event for E? (2 marks)
Ans: S = {10, 11, 12, 13, 14, 15, 16, 17}
- E (All even numbers) = {10, 12, 14, 16}
- E’ (complementary of E) = {11, 13, 15, 17}
Ques: Consider a three-coin toss where all outcomes are tails. What type of event is this? (2 marks)
Ans: Sample space for the coin toss will be,
- S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
- For the event A,
- A = {TTT}
- This event is only mapped to one element of sample space.
- Thus, it is a simple event.
Ques. You rolled a fair dice. What is the probability you get a 4 or a 3? (2 marks)
Ans: When we roll a die,
- Sample space, S = {1, 2, 3, 4, 5, 6}
- 4 or 3 denotes 4 and also 3
- Probability of getting 4 or 3 = \(\frac{2}{6}\) or \(\frac{1}{3}\)
For Latest Updates on Upcoming Board Exams, Click Here: https://t.me/class_10_12_board_updates
Check-Out:
Comments