CBSE Class 9 Maths Ch15 Probability Notes

Have you ever wondered how a cricket captain decides who should bowl the last over, or why we say there’s a “50-50 chance” of rain? That’s exactly where Probability comes in. From tossing a coin to predicting outcomes in games and daily situations, probability helps us make logical guesses instead of blind assumptions.

In CBSE Class 9 Maths Syllabus, Probability introduces you to the idea of how likely an event is to happen. You learn how to find outcomes, calculate chances and understand why some events are more likely than others.

Probability Notes Class 9

Stop flipping pages again and again trying to find the right points to study. We know how confusing it gets when exams are near and your book is full of long explanations. That’s why you’re here and that’s exactly why these Class 9 Probability notes are made for you.

Probability is actually a fun and practical chapter. It talks about chances, outcomes and everyday situations like tossing a coin or picking a card. But without proper notes, even simple topics can feel messy. These notes bring everything together in one place for you.

1. Introduction to Probability

Probability It refers to the measure of the chances of a certain event. We humans use probability in our daily life to predict future moments in our life.

Examples: In forecast, insurance, share market we use probability to predict future moment.

2. Important Definitions

Let us first familiarize ourselves with the terms that you shall get to see frequently in this chapter.

• Experiment: An experiment is referred to as random if it is being conducted without our knowledge of what will happen as the experiment's next result.

• Trial: A trial is an action that could have one or more outcomes, not just one. Take the spade 7 card out of the deck of cards, for instance, or determine the result of a dice roll, etc.

• Independent Trial: If a trial does not affect the outcome of any other random experiment, it is considered independent. Tossing a coin or rolling a die are examples of separate trials as they don't always affect one another.

• Event: The collection of experiment results is an event that occurs during experimentation. When we roll the dice, for instance, there is a three-percent chance that the result will be an odd number, such as 1, 3, or 5. So there would be three possible outcomes from the occurrence.

• Impossible Events: In a test, an event's probability is zero (0) if it is not conceivable for it to occur. We refer to this as an Impossible Event. For instance, you are not allowed to toss dice that have the number eight on them. Therefore, there is no chance in hell of rolling an 8 on a die.

• Sure or Certain Event: When an event is certain to occur during testing, it's referred to as having a particular probability. In this case, the likelihood is 1.  For instance, drawing a red ball from a bag containing exclusively red balls is guaranteed. This demonstrates that an event's probability could range from 1 to 0. Thus, P (E) ≤ 1 ≤ 0 ≤.

• Elementary Event: If there is only one possible outcome of an event to take place, then it is what we call an Elementary Event. For example, if we add all the elementary events of an experiment then their sum will be 1.

The general form is as given:

• P (H) + P (T) = 1
• P (H) + P= 1 (where H- ‘not H’).
• P (H) – 1 = P
• P (H) and P are called complementary events.

3. Formulae for Probability

Probability = Probability of an event = (Number of favorable outcomes) / (Total number of possible outcomes)

• Probability of an event is referred as P(E)
• The value of probability will always  lie between 0 and 1.
• If no outcome is favorable the probability will be 0
• If all outcomes are favorable the Probability  will be 1

Examples: 

Let us solve some examples now.

a) Tossing a Coin

In tossing a coin, there are always two phases: first is head and second is  tail. 

• Total outcomes = 2
• P(Head) = 1/2
• P(Tail) =1/2 

b) Rolling a Die

In a normal a die there are 1,2,3,4,5,6 numbers are availabe.

• In this case Total outcomes are  = 6
• Probability of getting:
• An even number (2, 4, 6)
• P(E)=3/6=1/2

4. Concept of Cards

To understand probability clearly, it is important to first know the concept of playing cards.

A standard pack of cards contains 52 cards in total. These cards are divided into four types (suits) and each suit has 13 cards.

Types of Cards and Their Details:

• Spades →13 cards, colour: Black
• Clubs → 13 cards, colour: Black
• Hearts → 13 cards, colour: Red
• Diamonds → 13 cards, colour: Red

So, the total number of cards in a pack is:

13 cards × 4 suits = 52 cards

Examples: 

Let us solve more examples based off probability.

Q1. Out of 35 students participating in a debate, 10 are girls. The Probability that the winner is a boy is: 

(a) 1/2 

(b) 7/3 

(c) 5/7

Ans. 5/7

Q2.What is the empirical probability that an engineer lives: 

(i) less than 7 km from her place of work? 

Solution: Total number of engineers =40 

Number of engineers living less than 7 km from her place of work= 9 

P(Engineer living less than 7 km from her place of work) 9/40

Q3: In a lottery box, there are 10 prizes and 25 blanks. A slip is drawn at random from the lottery box. What is the probability of getting a prize?

Solution:

Total number of prize = 10
Total number of blanks = 25

So, the total number of possible outcomes(i.e., n(S)) are = 10 + 25 = 35

According to the formula
Probability of getting a prize: P(E) = n(E)/n(S) = 1035 = 27

Q4: Cards and unmarried workers numbered 1 to 20 are mixed up and then a card is drawn at random. What is the probability that the card drawn has a number which is a multiple of 3 or 5?

Solution:

Cards are numbered from 1 to 20 therefore n(S) = {1, 2, 3, 4, ...., 19, 20}.

Let us consider E be the event of getting a multiple of 3 or 5
So, n(E) = {3, 6, 9, 12, 15, 18, 5, 10, 20}.

According to the formula
P(E) = n(E)/n(S) = 9/20.

Q5: One card is drawn from a deck of 52 cards, well-shuffled. Calculate the probability that the card will

(i) be an ace,
(ii) not be an ace.

Solution: 

Well-shuffling ensures equally likely outcomes.

(i) There are 4 aces in a deck.
Let us consider E to be the event the card drawn is ace.
The number of favorable outcomes to the event E = 4
The number of possible outcomes = 52

Therefore, P(E) = 4/52 = 1/13

(ii) Let us consider F be the event of ‘card is not an ace’
The number of favorable outcomes to F = 52 – 4 = 48
The number of possible outcomes = 52

Therefore, P(F) = 48/52 = 12/13

Q6: In a simultaneous throw of a pair of dice. Find the probability of getting a total of more than 7.

Solution:

Total number of combinations for a pair of dice is = n(S) = (6 x 6) = 36
Let us consider E be the event of getting a total more than 7
= {(2, 6), (3, 5), (3, 6), (4, 4), (4, 5), (4, 6), (5, 3), (5, 4),
(5, 5), (5, 6), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

Therefore, P(E) = n(E)/n(S)
= 15/36 = 5/12.

Q7: In a company of 364 workers, 91 are married. Find the probability of selecting an unmarried worker.

Solution: 

Given,
Total workers (i.e. Sample space) = n(S) = 364
Total married workers = 91

Now, total workers who are unmarried = n(E) = 364 – 91 = 273

Method 1: So, the probability of unmarried worker P(NM) = n(E)/n(S) = 273/364 = 0.75
Method 2: P(M) + P(NM) = 1
Here, P(M) = 91/364 = 0.25

So, 0.25 + P(NM) = 1
P(NM) = 1 – 0.25 = 0.75

Q8: From a bag of yellow and brown balls, the probability of picking a red ball is x/2. Find “x” if the probability of picking a brown ball is 2/3.

Solution:

Given, in the bag only yellow and brown balls.
P(picking a yellow ball) + P(picking a brown ball) = 1

x/2 + 2/3 = 1
3x + 4 = 6
3x = 2
Or, x = 2/3

In conclusion, these quick revision notes on Probability are designed to save your time and reduce last-minute stress. Instead of opening the textbook again and again, you can revise the entire chapter in just a few minutes using these clear and well-organized points.

These notes help you recall important concepts, formulas and ideas quickly, making revision smooth and effective. Go through them regularly, you will stay confident and one step ahead during exams.

FAQs

Q1. What is probability?

Ans. Probability is a branch of mathematics that deals with the chances of an event happening.

Q2. Why is probability studied in Class 9?

Ans. Probability is studied to understand real-life situations involving chance and to develop logical and analytical thinking.

Q3. What is an experiment in probability?

Ans. An experiment is an action or process that results in one or more possible outcomes, such as tossing a coin or rolling a dice.

Q4. What is an outcome in probability?

Ans. An outcome is a possible result of a random experiment.

Q5. How is probability useful in daily life?

Ans. Probability helps us predict results, make better decisions and understand uncertain situations like games and weather forecasts.