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I. Chapter Summary:

This chapter introduces the concept of theoretical (classical) probability and focuses on computing the likelihood of an event happening in simple experiments like tossing coins, rolling dice, or drawing cards. It aims to develop logical reasoning and decision-making skills. The emphasis is on calculating the probability of single events with equally likely outcomes in finite sample spaces.

II. Key Concepts Covered:

Concept	Explanation
Experiment	An action that produces outcomes (e.g., tossing a coin).
Outcome	A possible result of an experiment.
Sample Space	The set of all possible outcomes.
Event	A subset of the sample space (e.g., getting a head).
Probability of an Event (P(E))
$P(E) = frac{text{Number of favourable outcomes}}{text{Total number of outcomes}}$​
Valid only for equally likely outcomes.
Sure Event	Probability = 1 (event always occurs).
Impossible Event	Probability = 0 (event never occurs).
Complement of an Event (E’)	Probability that E does not occur:
$P(E’) = 1 – P(E)$

III. Important Questions:

(A) Multiple Choice Questions (1 Mark):

1. The probability of getting a prime number on a die is:
   a) $frac{1}{6}$
   b) $frac{2}{3}$
   c) $frac{1}{2}$ ✅
   d)$frac{5}{6}$
   (PYQ 2021)
2. A card is drawn from a well-shuffled pack of 52 cards. What is the probability of getting a king?
   a) $frac{1}{13}$ ✅
   b)$frac{1}{4}$
   c)$frac{1}{12}$
   d)$frac{4}{13}$
3. A coin is tossed once. What is the probability of getting a tail?
   a) 0
   b) 1
   c) $frac{1}{2}$✅
   d)$frac{1}{4}$
4. What is the probability of drawing a red queen from a deck of 52 cards?
   a) $frac{1}{13}$
   b) $frac{1}{26}$ ✅
   c) $frac{1}{52}$
   d) $frac{2}{13}$

(B) Short Answer Questions (2/3 Marks):

1. A number is selected at random from the numbers 1 to 100. What is the probability that it is divisible by 5?
2. Two coins are tossed simultaneously. Find the probability of getting:
   a) At least one head
   b) Both heads
3. A card is drawn at random. Find the probability that it is:
   a) A face card
   b) Not a heart
4. From numbers 1 to 20, find the probability that the number is:
   a) Prime
   b) A multiple of 3
   (PYQ 2022)

(C) Long Answer Questions (5 Marks):

1. A box contains 5 red balls, 3 blue balls, and 2 green balls. A ball is drawn at random. Find the probability that it is:
   a) Red
   b) Not blue
   c) Green
   d) Not red
   e) Either red or green
2. A card is drawn from a deck of 52 cards. What is the probability of getting:
   a) A red king
   b) A black queen
   c) A number card
   d) A card that is neither spade nor heart
   e) A joker (justify your answer)
3. A die is thrown once. Find the probability of getting:
   a) A number less than 4
   b) An even number
   c) A composite number
   d) A number not greater than 2
   e) An odd prime number
4. In a group of 40 students, 25 like cricket, 15 like football, and 10 like both. If a student is selected at random, find the probability that the student likes:
   a) Only cricket
   b) Only football
   c) Both
   d) At least one game
   e) Neither game

(D) HOTS (Higher Order Thinking Skills):

1. A bag contains 10 balls numbered 1 to 10. A ball is drawn at random and not replaced. Another ball is drawn. Find the probability that:
   a) Both balls are even numbers
   b) One ball is even and the other is odd
2. A letter is chosen at random from the word “MATHEMATICS”. What is the probability that the letter is:
   a) A vowel
   b) A consonant
   c) Not a repeated letter

IV. Key Formulas/Concepts:

Concept	Formula/Definition
Probability of an Event E	$P(E) = frac{text{Number of favourable outcomes}}{text{Total number of outcomes}}$
Probability of Sure Event	$P(E) = 1$
Probability of Impossible Event	$P(E) = 0$
Complementary Event	$P(E’) = 1 – P(E)$
Total Probability Range	$0 leq P(E) leq 1$

V. Deleted Portions (CBSE 2025–2026):

No portions have been deleted from this chapter as per the rationalized NCERT textbooks.

VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025–26):

Unit/Chapter	Estimated Marks	Type of Questions Typically Asked
Probability	3–4 Marks	1 Short Answer + 1 MCQ or HOTS-based

VII. Previous Year Questions (PYQs):

Year	Marks	Question
2021	1M	Probability of prime number on a die
2022	3M	Random number from 1 to 20, find probability of event
2020	3M	Drawing colored balls from a bag
2019	5M	Probability of events in a deck of cards (multiple subparts)

VIII. Real-World Application Examples:

Scenario	Concept Used
Weather forecasting	Probability of rain
Insurance & Risk Management	Life expectancy, accident probability
Gaming & Sports	Winning chances, betting odds
Genetics	Probability of inheriting traits
Elections	Predicting winning chances

IX. Student Tips & Strategies for Success (Class-Specific):

 Time Management:

• Spend 15 minutes/day on probability problems.
• Use real-life scenarios to practice quick estimation.

 Exam Preparation:

• Write the total number of outcomes first before calculating favourable ones.
• Always simplify fractions and verify that probabilities lie between 0 and 1.
• Use Venn diagrams to visualize mutually exclusive events.

 Stress Management:

• Practice with fun problems (like card games, coins, dice).
• Don’t overthink simple problems; follow the basic formula.

X. Career Guidance & Exploration (Class-Specific):

For Classes 9–10:

Stream	Career Paths	Related Exams
Science	Data Analyst, AI Engineer, Mathematician	NTSE, Olympiads
Commerce	Actuary, Risk Manager, Investment Banker	NSE, CUET
Arts	Political Analyst, Journalist (surveys/stats)	Aptitude-based exams

Tip: Strong understanding of probability aids in logical reasoning sections of JEE, CUET, NDA, SSC, and more.

XI. Important Notes:

•  Refer to the official https://cbseacademic.nic.in and https://ncert.nic.in for syllabus updates.
•  Always express answers in lowest fractional form unless asked for decimal/percentage.
•  Probability builds critical thinking – key for competitive exams and real-world decisions.
•  Practice different types of problems — single event, combined event, complementary events.