CBSE Class 9 Maths — Probability

Chapter Overview & Weightage

Probability in Class 9 is the empirical (experimental) probability — based on actual outcomes from experiments, not theoretical formulae. This chapter introduces the vocabulary of probability and the concept that probability is a ratio between 0 and 1.

In CBSE Class 9 board exams, Probability carries 4–6 marks — typically one 1-mark MCQ and one 3-mark or 4-mark problem. The chapter is purely application-based: read carefully, identify favourable outcomes, and divide by total outcomes.

Year	Marks Allotted	Question Type
2023	4 marks	1 MCQ + 1 Long Answer
2022	4 marks	1 MCQ + 1 Short Answer
2021	3 marks	1 Short Answer
2020	5 marks	1 MCQ + 1 Long Answer

The chapter is relatively easy scoring — most students who understand the basic formula get full marks.

Key Concepts You Must Know

1. Experiment: Any activity with a well-defined set of outcomes. Rolling a die, tossing a coin.

2. Trial: One performance of the experiment.

3. Event: A collection of outcomes we are interested in.

4. Favourable outcome: An outcome that satisfies the event condition.

5. Empirical Probability Formula:

P(E) = \frac{\text{Number of trials in which event E occurred}}{\text{Total number of trials}}

6. Properties of Probability:

$0 \leq P(E) \leq 1$ always
$P(E) + P(\bar{E}) = 1$ , where $\bar{E}$ is the event “E does not occur”
$P(\text{impossible event}) = 0$
$P(\text{certain event}) = 1$

Important Formulas

P(E) = \frac{\text{Number of favourable outcomes (in observed data)}}{\text{Total number of trials (observations)}}

P(\bar{E}) = 1 - P(E)

If probability of winning is 0.3, probability of not winning = 0.7.

Solved Previous Year Questions

PYQ 1 — CBSE 2023 (3 marks)

A die is thrown 500 times. The outcomes are recorded:

Outcome	1	2	3	4	5	6
Frequency	80	75	90	95	85	75

Find the probability of getting: (i) an even number, (ii) a number greater than 4.

Solution:

Total trials = 500

(i) Even numbers: 2, 4, 6. Frequency = 75 + 95 + 75 = 245

P(\text{even}) = \frac{245}{500} = \frac{49}{100} = 0.49

(ii) Numbers > 4: 5 and 6. Frequency = 85 + 75 = 160

P(\text{number} > 4) = \frac{160}{500} = \frac{8}{25} = 0.32

PYQ 2 — CBSE 2022 (4 marks)

Out of 100 students, 60 are boys. A student is chosen at random. Find: (i) P(student is a girl), (ii) P(student is a boy).

Solution:

Total students = 100. Boys = 60, Girls = 40.

(i) $P(\text{girl}) = \frac{40}{100} = \frac{2}{5} = 0.4$

(ii) $P(\text{boy}) = \frac{60}{100} = \frac{3}{5} = 0.6$

Verify: $P(\text{boy}) + P(\text{girl}) = 0.6 + 0.4 = 1$ ✓

PYQ 3 — CBSE Level (Application)

A bag has 3 red, 5 blue, and 2 green balls. A ball is drawn at random. Find the probability it is (i) red, (ii) not blue.

Solution:

Total balls = 10.

(i) $P(\text{red}) = \frac{3}{10}$

(ii) $P(\text{not blue}) = 1 - P(\text{blue}) = 1 - \frac{5}{10} = \frac{1}{2}$

Difficulty Distribution

For CBSE Class 9 Probability:

Difficulty	Approximate % of Questions
Easy (direct formula application)	60%
Medium (multi-part with complement)	30%
Hard (frequency table + multiple events)	10%

This is one of the more scoring chapters — consistent practice of 10–15 questions ensures full marks.

Expert Strategy

Step 1: Read the data carefully. Most errors happen from misreading the table or miscounting totals. Before calculating, write down: Total trials = ?, Favourable outcomes for the event = ?

Step 2: Simplify the fraction. CBSE expects the probability in simplest form. $\frac{245}{500}$ should be simplified to $\frac{49}{100}$ .

Step 3: Verify using complement. For multi-part questions, the sum of all probabilities = 1. Use this to cross-check: $P(\text{head}) + P(\text{tail}) = 1$ .

Step 4: Express as fraction and decimal. Board papers often ask for probability in both forms — write both unless the question specifies.

The complement rule $P(\bar{E}) = 1 - P(E)$ is your fastest tool for “not happening” questions. If you’re asked for the probability of getting “at least one” in a sequence, it’s much easier to compute $1 - P(\text{none})$ .

Common Traps

Trap 1: Dividing by the wrong total. If a frequency table shows 6 categories with frequencies 80, 75, 90, 95, 85, 75, the total is their sum (500), not 6 (the number of categories). Always add up the frequencies.

Trap 2: Writing probability outside [0,1]. If you get P(E) = 1.2 or P(E) = -0.3, something is wrong. Probability is always between 0 and 1 inclusive. Check your calculation.

Trap 3: Confusing empirical and theoretical probability. In Class 9, we only use empirical probability — based on actual recorded data. “A coin is tossed 100 times; 55 heads observed” → P(head) = 55/100, not 1/2. Class 10 introduces theoretical probability where we use 1/2 for a fair coin.

Trap 4: Not simplifying the fraction. CBSE deducts marks for unsimplified fractions. Always check for common factors in numerator and denominator.