← Unit 4 Two-Stage Events and Tree Diagrams

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Two-Stage Events and Tree Diagrams

⏱ 25 min📚 Year 9📈
Think First

A bag has 2 red and 3 blue marbles. You draw one, replace it, then draw again. What is P(two reds)?

💡 Revision: Ensure you understand basic probability and can multiply fractions.

Learning Intentions

Know

  • Two-stage experiment
  • Tree diagram
  • With replacement
  • Without replacement
  • Multiplication rule

Understand

  • Why replacement affects probabilities
  • How tree diagrams organise outcomes

Can Do

  • Draw tree diagrams
  • Calculate probabilities for with/without replacement
  • Use the multiplication rule for independent events
Tree diagramWith replacementWithout replacementIndependentDependent
Learn Phase
1

Two-Stage Experiments

Experiments with two parts

A two-stage experiment involves two actions, such as drawing two cards or tossing two coins.

The probability of a specific sequence is found using:

Multiplication Rule

$P(A ext{ then } B) = P(A) imes P(B ext{ after } A)$

Whether you replace the first item affects the second probability.

2

With Replacement

Independent events

When the first item is replaced, the second draw is independent — the probabilities don't change.

Example: Bag with 3 red, 2 blue. Draw, replace, draw.

P(red then red) = $ rac{3}{5} imes rac{3}{5} = rac{9}{25}$

Tree diagram has identical branches at each stage.

3

Without Replacement

Dependent events

When the first item is not replaced, the second draw is dependent — the probabilities change.

Example: Bag with 3 red, 2 blue. Draw, keep out, draw.

P(red then red) = $ rac{3}{5} imes rac{2}{4} = rac{6}{20} = rac{3}{10}$

Tree diagram branches change after the first stage.

✓

Check Understanding

Try it yourself

A bag has 4 red and 6 blue marbles. Two are drawn without replacement. Find P(both red) and P(one of each colour).

Worked Example

Two-Stage Probability

1

A coin is tossed twice. Draw a tree diagram and find P(two heads) and P(exactly one tail).

Tree: H-H, H-T, T-H, T-T (each with prob 1/4)

P(HH) = 1/4

P(exactly one T) = P(HT) + P(TH) = 1/4 + 1/4 = 1/2

2

A bag has 5 red, 5 blue. Draw two with replacement. Find P(both red) and P(different colours).

P(RR) = $ rac{5}{10} imes rac{5}{10} = rac{25}{100} = rac{1}{4}$

P(different) = P(RB) + P(BR) = $ rac{25}{100} + rac{25}{100} = rac{1}{2}$

3

Same bag, without replacement. Find P(both red).

P(RR) = $ rac{5}{10} imes rac{4}{9} = rac{20}{90} = rac{2}{9}$

Common Misconceptions

✕

Treat without replacement as with replacement. This is a very common error. Without replacement, the denominator decreases and probabilities change.

✕

Add probabilities along branches instead of multiplying. You multiply along branches (AND) and add across branches (OR).

✕

Forget that order matters for "one of each." P(one red, one blue) = P(RB) + P(BR), not just one branch.

Your Turn

Practice — Tree Diagrams

Work through each question in your book or digitally. Answers are in the Questions phase.

1A bag has 3 green, 2 yellow marbles. Two drawn with replacement. Draw tree and find P(both green).
2Same bag, without replacement. Find P(both green) and P(one of each).
3A die is rolled, then a coin tossed. Find P(6 and heads).
Real-World Anchor

Genetics and Medicine

Genetic inheritance involves two-stage probability. The Punnett square is essentially a tree diagram showing probabilities of inheriting traits from parents. Medical testing uses sequential probability — the probability of disease given a positive test depends on whether the test was done with or without prior screening (replacement analogy).

📓 Copy Into Your Books

▼

Multiplication rule

  • P(A then B) = P(A) × P(B after A)
  • Multiply along branches

With replacement

  • Independent
  • Probabilities stay the same

Without replacement

  • Dependent
  • Denominator decreases
  • Probabilities change
Questions Phase
Check Your Understanding
Answer all questions correctly to unlock the Game phase.
With replacement, events are:
Bag: 2R, 3B. With replacement. P(RR) =
Same bag, without replacement. P(RR) =
Tree diagrams: you multiply along branches for:
Coin tossed twice. P(exactly one head) =
Bag: 4R, 4B. Without replacement. P(different colours) =
P(A and B) for independent events =
Without replacement, second draw probability:
1A bag has 6 red, 4 blue marbles. Two drawn with replacement. Find P(both blue) and P(at least one red).
2Same bag, without replacement. Find P(both blue) and P(one of each).
3Explain the difference between independent and dependent events with examples.

Comprehensive Answers

1With replacement: P(both blue), P(at least one red).
P(BB)=(4/10)²=16/100=4/25. P(at least one red)=1-P(BB)=21/25.
2Without replacement: P(both blue), P(one each).
P(BB)=4/10×3/9=12/90=2/15. P(one each)=6/10×4/9+4/10×6/9=48/90=8/15.
3Independent vs dependent.
Independent: one event does not affect the other (with replacement). Dependent: first affects second (without replacement).
MC 1With replacement.
Independent. Answer: B
MC 2P(RR) with replacement.
(2/5)²=4/25. Answer: A
MC 3P(RR) without replacement.
2/5×1/4=2/20=1/10. Answer: B
MC 4Multiply along branches.
AND. Answer: B
MC 5P(exactly one head).
P(HT)+P(TH)=1/2. Answer: B
MC 6P(different) without replacement.
4/8×4/7+4/8×4/7=32/56=4/7. Answer: C
MC 7P(A and B) independent.
P(A)×P(B). Answer: B
MC 8Without replacement second draw.
Changes. Answer: B
SA 1With replacement.
P(BB)=4/25, P(at least one red)=21/25.
SA 2Without replacement.
P(BB)=2/15, P(one each)=8/15.
SA 3Independent vs dependent.
With replacement = independent; without = dependent.
Game Phase
🎲
Game Unlocked!
You have mastered the Check Your Understanding questions. Choose a game mode below.
📦
Classify & Sort
Sort mathematical objects by their properties.
⚡
Speed Challenge
Answer questions against the clock.
📈
Match Maker
Match problems to their solutions.