Probability

Complete Class 10 Probability Notes for Quick Learning

Class 10 Probability Notes provide a complete guide to understanding simple probability, events, and probability calculations with step-by-step examples. These notes focus on practical applications, solving problems efficiently, and building a strong foundation for higher mathematics.

Understanding probability is essential for exams, real-life scenarios, and analytical problem-solving. The notes include tips, solved exercises, and shortcuts to calculate probability quickly. With structured practice questions, students can strengthen their reasoning skills, improve accuracy, and confidently attempt probability-related questions in board exams. These notes are perfect for Class 10 students looking for clear explanations, effective revision, and success in mathematics. Students must solve the NCERT questions with the help of the NCERT Solutions for class 10 Maths, which will help in building the foundation.

Class 10 Probability Notes, Solved example & Questions

Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth is not certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The certainty we adopt can be described in terms of a numerical measure and this number, between 0 and 1, we call probability. The higher the probability of an event, the more certain we are that the event will occur. Thus, probability in an applied sense is a measure of the confidence a person has that a (random) event will occur.

Experiment

The word experiment means an operation, which can produce well defined outcomes. There are two types of experiment:

1. Deterministic experiment
2. Probabilistic or Random experiment

Deterministic Experiment:

Those experiments which when repeated under identical condition produced the same results or outcome are known as deterministic experiment. For example, Physics or Chemistry experiments performed under identical conditions.

Probabilistic Or Random Experiment:

In an experiment, when repeated under identical conditions do not produce the same outcomes every time. For example, in tossing a coin, one is not sure that if a head or tail will be obtained. So it is a random experiment.

Sample Space:

The set of all possible outcomes of a random experiment is called a sample space associated with it and is generally denoted by S. Example: When a dice is tossed then S = {1, 2, 3, 4, 5, 6}.

Event

A subset of sample space associated with a random experiment is called an event. For example, in tossing a dice getting an even number is an event.

Also Read: Arithmetic Progressions Notes

Favourable Event:

Let S be a sample space associated with a random experiment and A be event associated with the random experiment. The elementary events belonging to A are known as favourable events to the event A. For example, in throwing a pair of dice, A is defined by "Getting 8 as the sum". Then following elementary events are as outcomes: (2, 6); (3, 5); (4, 4); (5, 3); (6, 2). So, there are 5 elementary events favourable to event A.

Complementary Event:

Let E be an event and (not E) be an event which occurs only when E does not occur. The event (not E) is called the complementary event of E. Clearly, P(E) + P(not E) = 1. ∴ P(E) = 1 – P(not E).

Probability

If there are n elementary events associated with a random experiment and m of them are favourable to an event A, then the probability of happening or occurrence of event A is denoted by P(A) P(A) = m/n = Total number of favourable outcomes / Total number of possible outcomes And 0 ≤ P(A) ≤ 1

• If P(A) = 0, then A is called impossible event
• If P(A) = 1, then A is called sure event
• P(A) + P(Ā) = 1
• Where P(A) = probability of occurrence of A, P(Ā) = probability of non-occurrence of A.

Some Special Sample Spaces

Experiment	Sample Space (S)	Number of Outcomes n(S)
A die is thrown once	S = {1,2,3,4,5,6}	n(S) = 6
A coin is tossed once	S = {H, T}	n(S) = 2
A coin is tossed twice or Two coins are tossed simultaneously	S = {HH, HT, TH, TT}	n(S) = 4 = 2²
A coin is tossed three times or Three coins are tossed simultaneously	S = {HHH, HHT, HTH, THH, TTT, TTH, THT, HTT}	n(S) = 8 = 2³
Two dice are thrown together or A die is thrown twice	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)}	n(S) = 6²

Solved Examples

Question: A die is thrown once. What is the probability of getting a prime number?

Solution: In a single throw of a die, all possible outcomes are 1, 2, 3, 4, 5, 6. Total number of possible outcomes = 6.

Let E be the event of getting a prime number. Then, the favourable outcomes are 2, 3, 5. Number of favourable outcomes = 3.

∴ P(getting a prime number) = P(E) = 3/6 = 1/2.

Must Read: Quadratic Equations Notes

Question: A coin is tossed once. What is the probability of getting a head?

Solution: When a coin is tossed once, all possible outcomes are H and T.

Total number of possible outcomes = 2.

The favourable outcome is H.

Number of favourable outcome = 1

∴ P(getting a head) = P(H) = 1/2.

Question: A bag contains a red ball, a blue ball and a yellow ball, all the balls being of the same size. Kritika takes out a ball from the bag without looking into it. What is the probability that she takes out the

(i) Yellow ball? (ii) Red ball? (iii) Blue ball?

Solution: Kritika takes out a ball from the bag without looking into it. So, it is equally likely that she takes out any one of them. Let Y be the event 'the ball taken out is yellow', B be the event 'the ball taken out is blue', and R be the event 'the ball taken out is red'. Now, the number of possible outcomes = 3.

(i) The number of outcomes favourable to the event Y = 1. So, P(Y) = 1/3 Similarly,

(ii) P(R) = 1/3 and

(iii) P(B) = 1/3

Also Read: Statistics Notes

Question: All the three face cards of spades are removed from a well-shuffled pack of 52 cards. A card is then drawn at random from the remaining pack. Find the probability of getting.

(i) Black face card

(ii) a queen

(iii) a black card.

Solution: After removing three face cards of spades (king, queen, and jack) from a deck of 52 playing cards, there are 49 cards left in the pack. Out of these 49 cards one card can be chosen in 49 ways.

Total number of elementary events = 49

(i) There are 6 black face cards out of which 3 face cards of spades are already removed. So, out of remaining 3 black face cards one black face card can be chosen in 3 ways.

Favourable number of elementary events = 3 Hence, P(getting a black face card) = 3/49

(ii) There are 3 queens in the remaining 49 cards.

So, out of these three queens, one queen can be chosen in 3 ways

Favourable number of elementary events = 3

Hence, P(Getting a queen) = 3/49 (iii) There are 23 black cards in the remaining 49 cards. So, out of these 23 black cards, one black card can be chosen in 23 ways

Favourable number of elementary events = 23

Hence, P(Getting a black card) = 23/49

Question: A die is thrown, Find the probability of

(i) prime number

(ii) multiple of 2 or 3

(iii) a number greater than 3

Solution: In a single throw of die any one of six numbers 1,2,3,4,5,6 can be obtained. Therefore, the total number of elementary events associated with the random experiment of throwing a die is 6.

(i) Let A denote the event "Getting a prime no". Clearly, event A occurs if any one of 2,3,5 comes as outcome.

Favorable number of elementary events = 3

Hence, P(Getting a prime no.) = 3/6 = 1/2 (ii) A multiple of 2 or 3 is obtained if we obtain one of the numbers 2,3,4,6 as outcomes Favorable number of elementary events = 4

Hence, P(Getting multiple of 2 or 3) = 4/6 = 2/3

(iii) The event "Getting a number greater than 3" will occur, if we obtain one of number 4,5,6 as an outcome. Favorable number of outcomes = 3 Hence, required probability = 3/6 = 1/2

EXERCISE - 1

1. If three coins are tossed simultaneously, then the probability of getting at least two heads, is (A) 1/4 (B) 3/8 (C) 1/2 (D) 1/4
2. A bag contains three green marbles four blue marbles, and two orange marbles. If marble is picked at random, then the probability that it is not a orange marble is (A) 1/4 (B) 1/3 (C) 4/9 (D) 7/9
3. A number is selected from number 1 to 27. The probability that it is prime is (A) 2/3 (B) 1/6 (C) 1/3 (D) 2/9
4. If P(E) = 0.05, then P(not E) = (A) -0.05 (B) 0.5 (C) 0.9 (D) 0.95
5. A bulb is taken out at random from a box of 600 electric bulbs that contains 12 defective bulbs. Then the probability of a non-defective bulb is (A) 0.02 (B) 0.98 (C) 0.50 (D) None

Also Read: Probability Notes

Answers to Exercise - 1:

1. (C) 2. (D) 3. (C) 4. (D) 5. (B)

Important Points To Remember

Summary

Probability is a fundamental concept in mathematics that helps us quantify uncertainty. Understanding the basic concepts of sample space, events, and the probability formula is essential for solving problems in this area. Regular practice with various types of problems involving coins, dice, cards, and other scenarios will help build confidence and proficiency in probability calculations.