GANITA MANJARI · GRADE 9 · PART I · CHAPTER 7
Study Materials
Think first. Use AI purposefully. Verify always.
CHANCE — Uncertainty and random experiments
A random experiment has identifiable possible outcomes, but its next result is uncertain under the model. A probability describes likelihood, not a promise about a single trial.
Describe the experiment and what is recorded. List possible results. State assumptions such as a fair coin, random toss and independent repeats before assigning probabilities.
Two possible outcomes alone do not make them equally likely. A car starting and not starting need not have equal chances.
Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 7 · printed page 155
Worked example →SCALE — Probability from 0 to 1
Probability lies from 0 to 1. In these finite models, 0 is impossible and 1 is certain. At an event and its complement have equal chances. The complement contains all outcomes outside the event.
Name the event and its complement. Locate the probability between 0 and 1. Use P(not E) = 1 − P(E). A decimal can also be written as a fraction or percentage.
If a match can be drawn, P(win) = means P(not win) = ; it does not by itself mean P(lose) = .
Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 7 · printed page 157
Worked example →OBSERVE — Experimental probability and relative frequency
Experimental probability is the observed event count divided by the positive number of trials. It reports this record; different records can give different proportions.
Use a positive trial count. This is an observed ratio.
Record every trial consistently. Count occurrences of the specified event, divide by the number of trials and label the answer experimental. With zero trials the ratio is not yet defined.
An observed frequency is not a guarantee about the next trial or evidence that the exact model probability equals that frequency.
Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 7 · printed page 160
Worked example →EXPECT — Theoretical probability and equal likelihood
For a finite model with equally likely elementary outcomes, P(E) = n(S). The elementary outcomes must retain distinctions that make selection equally likely, such as separately labelled cards.
Choose an elementary sample space. Justify equal likelihood. Identify the event, count its outcomes, divide by n(S) and simplify. If outcomes have unequal probabilities, count alone is insufficient.
The colour outcomes {red, blue} are not equally likely when there are 3 red and 7 blue equally selectable marbles.
Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 7 · printed page 161
Worked example →COMPARE — Experimental versus theoretical probability
The theoretical probability of heads in independent fair tosses is . Observed proportions fluctuate. In long runs they tend to be closer to the model probability; no fixed trial count guarantees exact equality or steadily improving closeness.
Keep the event and experiment the same. Compare the observed ratio with the theoretical value. Examine several runs and sample sizes instead of selecting one convenient record.
A longer run need not be closer at every step. A run of heads does not require an immediate balancing run of tails.
Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 7 · printed page 163
Worked example →SAMPLE — Evidence, samples and representativeness
A sample provides evidence about a population, not certainty. How people are selected matters. A larger sample drawn through a biased method can still be unrepresentative.
Separate the observed sample proportion from the population estimate. Describe selection and possible bias. Sample across the relevant groups and label extrapolations as estimates.
n(S), the number of possible outcomes, is not the number of people in a statistical survey. Neither a large sample nor an estimate guarantees the true population count.
Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 7 · printed page 162
Worked example →SPACE — Sample spaces, n(S) and events
S contains all possible recorded outcomes, without duplicates. n(S) counts its elements. An event E is a subset of S. Choose enough detail for the question; grouped outcomes need not be equally likely.
Specify what is recorded and how order is distinguished. Enumerate systematically. Check completeness and uniqueness, then select the outcomes satisfying the event.
Recording only the number of heads gives {0,1,2}, but these values are not equally likely for two independent fair tosses.
Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 7 · printed page 166
Worked example →BRANCH — Trees and multi-step experiments
A tree lists complete paths through sequential trials. Multiply conditional branch probabilities along a path. Add probabilities of disjoint paths for an event. Multi-step experiments can be independent or dependent.
Label each stage. After every outcome update the state if needed. Branch probabilities leaving a possible node must sum to 1. Keep the order of the draws.
Without replacement, the second denominator decreases and the count of the colour removed decreases. Using the original fractions again is incorrect.
Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 7 · printed page 168
Worked example →REASON — Fairness, dependence and careful conclusions
Past independent outcomes do not change the next trial’s model probability. Removing an object can change the next draw’s probabilities. Fairness, replacement and uniformity are assumptions that must be stated—not guessed.
Identify the model and its assumptions before calculating. Check whether the state changes. For a uniformly distributed landing point, compare areas rather than counting a finite list of points.
A geometric area ratio requires uniform landing over the whole region. Merely saying a point lands somewhere does not justify that assumption.
Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 7 · printed page 164
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