The probability of an impossible event is
- (a)\(0\)
- (b)\(\dfrac12\)
- (c)\(1\)
- (d)\(-1\)
Introduction to Probability Class 9: notes and important questions
Revision notes, 17 board-style questions with the step marks shown, and a four-step route from the basics to 95+, each step ending in a short checkpoint.
- 17 questions
- 5 multiple choice, 1 assertion–reason, 4 very short answer, 3 short answer, 2 long answer, 2 case study
- About 7 hours to master
Statistics and Probability unit: 10 of 80 theory marks (Statistics, Introduction to Probability).
In the NCERT book Ganita Manjari: Part I, Chapter 7, The Mathematics of Maybe: Introduction to Probability.
Revision notes
Introduction to Probability: revision notes
1. The probability scale
Probability measures how likely an event is, on a scale from \(0\) to \(1\): \(0\) for an impossible event, \(1\) for a certain event, \(\dfrac12\) for "as likely as not". For any event \(E\), \(P(\text{not } E) = 1 - P(E)\).
2. Experimental probability
From trials or recorded data: \(P(E) = \dfrac{\text{number of trials in which } E \text{ happened}}{\text{total number of trials}}\). It changes from one experiment to another and settles down as the number of trials grows. Expected number of times in \(N\) new trials \(\approx N \times P(E)\) (an estimate, not a promise).
3. Theoretical probability
When all outcomes are equally likely: \(P(E) = \dfrac{\text{number of outcomes favourable to } E}{\text{total number of outcomes}}\). A fair coin: \(P(H) = \dfrac12\); a fair die: each face \(\dfrac16\).
4. Sample spaces and events
The sample space is the list of all possible outcomes; an event is a set of outcomes. Two coins: \(\{HH, HT, TH, TT\}\); a coin and a die: \(12\) outcomes; two dice: \(36\) ordered pairs \((a, b)\).
5. Tree diagrams
For an experiment in stages, draw one set of branches per stage. Each path from the start is one outcome; with equally likely branches, count the paths. Three coins give \(2 \times 2 \times 2 = 8\) paths.
Worked example 1
A fair die is thrown. Find the probability of a number greater than \(4\).
Favourable \(\{5, 6\}\): \(P = \dfrac26 = \dfrac13\).
Worked example 2
In \(250\) tosses a coin showed heads \(135\) times. Find the experimental probability of a tail.
Tails \(= 115\), so \(P(T) = \dfrac{115}{250} = \dfrac{23}{50}\).
Worked example 3
Two fair coins are tossed. Find \(P(\text{exactly one head})\).
Outcomes \(HH, HT, TH, TT\); favourable \(HT, TH\): \(P = \dfrac12\).
Common errors
- Listing two coins as \(\{2H, 1H, 0H\}\): these are not equally likely; list \(HT\) and \(TH\) separately.
- Using the theoretical value when the question asks for the experimental probability from data.
- Probabilities greater than \(1\) or negative: check every answer lies in \([0, 1]\).
- Counting an outcome twice (for example \(15\) as both a multiple of \(3\) and of \(5\)).
Exam tips
- Write the sample space (or its size) and the favourable outcomes before the fraction.
- Give probabilities as fractions in lowest terms.
Topics in this chapter: The probability scale · Theoretical probability · Experimental probability · Sample spaces and events · Tree diagrams.
Route to 95: four steps
Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.
Step 1
Secure the basics
You can place events on the probability scale and find simple experimental and theoretical probabilities.
Step 2
Exam standard
You can list sample spaces for coins, dice and spinners, use data tables and estimate expected frequencies.
Step 3
Full marks on long answers
You can use tree diagrams and solve multi-part problems on bags, spinners and recorded data.
Step 4
95+ stretch (HOTS)
You can handle two-dice and two-spinner events such as 'the first number is larger' or 'sum greater than product'.
Practice questions
Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 2 of the 17 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.
A fair die is thrown once. The probability of getting a prime number is
- (a)\(\dfrac13\)
- (b)\(\dfrac16\)
- (c)\(\dfrac23\)
- (d)\(\dfrac12\)
A coin is tossed \(200\) times and shows heads \(116\) times. The experimental probability of getting a tail is
- (a)\(\dfrac{29}{50}\)
- (b)\(\dfrac12\)
- (c)\(\dfrac{21}{50}\)
- (d)\(\dfrac{84}{100}\)
Stretch yourself: original probability problems at Intermediate level (ages 13 to 16), with hints and full solutions: Problem I129 · Problem I136 · Problem I143 · Problem I34. All 26 →
Where marks are lost in Introduction to Probability
- Outcomes that are not equally likely used as the sample space (0, 1, 2 heads). Fix: list ordered outcomes HH, HT, TH, TT.
- Theoretical probability given when the data asked for the experimental one. Fix: use the counts from the table.
- Double counting in 'or' events (15 is a multiple of both 3 and 5). Fix: list the favourable outcomes.
- Expected frequency stated as certain. Fix: 'about 140', because experimental probability is an estimate.
- Fractions not simplified or probabilities outside 0 to 1. Fix: check the final answer against the scale.
Test Introduction to Probability against the clock
Want it against the clock? Take a timed 30-mark chapter test on Introduction to Probability, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Essentials or the free trial).