GANITA MANJARI · GRADE 9 · PART I · CHAPTER 7

BRANCH — Trees and multi-step experiments

Think first. Use AI purposefully. Verify always.

DISCOVER

Begin with your own prediction.

A red ball is drawn from four red and five blue balls and kept aside. How many balls remain for the second draw?

Write or sketch a first attempt. Use the model to test your assumptions before accepting a rule.

Understand

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.

Key Ideas

\frac{4}{9} \times \frac{5}{8} = \frac{5}{18}

Without replacement, the second denominator decreases and the count of the colour removed decreases. Using the original fractions again is incorrect.

Visualise

Follow the conditional branches

Experiment

Two independent fair tosses · independent trials. Read each route from Start to its final outcome.

Start

First: H · $\frac{1}{2}$

↳ Second: H · $\frac{1}{2}$
Outcome: HH
↳ Second: T · $\frac{1}{2}$
Outcome: HT

First: T · $\frac{1}{2}$

↳ Second: H · $\frac{1}{2}$
Outcome: TH
↳ Second: T · $\frac{1}{2}$
Outcome: TT

Multiply conditional probabilities along a path. Add disjoint path probabilities for a larger event. At every possible node, the outgoing probabilities sum to 1.

Method

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.

Worked Example

Given

4 red and 5 blue balls; two random draws without replacement (starred Q10).

To Find

Probability of red followed by blue

Reasoning

First draw: R has probability $\frac{4}{9}$ . After R: 3 red and 5 blue remain.

Solution

Favourable colour path: RB.
Probability calculation: ( $\frac{4}{9}$ ) × ( $\frac{5}{8}$ ) = $\frac{20}{72}$ = $\frac{5}{18}$ .

Result

P(RB) = $\frac{5}{18}$ .

AIMS Method

A — Attempt independently
Predict and describe what is being recorded.
I — Investigate the idea
Identify the model, evidence and assumptions.
M — Make connections
Connect outcomes, events, fractions and tree paths.
S — Scrutinise your reasoning
Check completeness, likelihoods and changing conditions.

Think first. Use AI purposefully. Verify always.

Focused Practice

Find the probability of two blue balls in the same no-replacement experiment.

Your reasoning

Continue with TREE DIAGRAMS →

Check & Feedback

Explain the model, state an assumption and identify one conclusion the information does not justify.

Explain your conclusion

Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 7 · printed page 168