Mathematics • Year 7 • Unit 4 • Lesson 19
Two-Stage Experiments — Mixed Challenge
Bring together with/without replacement, multiplication along tree branches, and complementary events (at least one). Spot a common mistake and design your own two-stage experiment.
1. Mixed problems
Show working. State "with" or "without" replacement first. 2 marks each
1.1 Bag: 7 red, 3 blue. Draw 2 WITH replacement. Find P(both blue).
1.2 Same bag (7 red, 3 blue), draw 2 WITHOUT replacement. Find P(both blue). Compare with 1.1.
1.3 A coin is flipped, then a die is rolled. Are these independent? Find P(heads AND rolling a 4).
1.4 Two cards from a 52-card deck without replacement. Use the complement rule to find P(at least one heart) = 1 − P(no hearts in 2 draws).
1.5 A team has 6 boys and 4 girls. Two players are picked at random WITHOUT replacement to lead the warm-up. Find P(one boy AND one girl, in any order).
1.6 A bag has 4 red and 6 blue lollies. Two are drawn WITHOUT replacement. (i) Find P(red then blue). (ii) Find P(blue then red). (iii) Briefly state why these two answers are equal even though the drawing order matters within each event.
2. Find the mistake
A Year 7 student is asked: "A bag has 5 red and 3 blue marbles. Two are drawn WITHOUT replacement. Find P(both red)." Exactly one line contains a serious error. Spot it, explain why it's wrong, then redo the calculation correctly. 3 marks
Student's working:
Line 1: Total marbles = 5 + 3 = 8.
Line 2: P(R on draw 1) = 5/8.
Line 3: P(R on draw 2) = 5/8 (pool unchanged).
Line 4: P(RR) = 5/8 × 5/8 = 25/64.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Show the correct second-draw probability and the correct final answer.
Stuck? Without replacement, after one red is removed, only 4 reds remain in 7 marbles.3. Open-ended challenge — design your own two-stage experiment
This question has many correct answers. Show your work clearly. 4 marks
3.1 Design a two-stage selection scenario from real school life. You must provide:
- a setup describing the pool (numbers, types) in 1–2 sentences,
- one question solved with replacement, with a worked multiplication and a final probability,
- the same question solved without replacement, with a worked multiplication and a final probability,
- a one-sentence comparison of the two answers.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — 7R, 3B, with replacement, both blue
P(BB) = 3/10 × 3/10 = 9/100 = 0.09.
1.2 — Same bag, without replacement, both blue
P(BB) = 3/10 × 2/9 = 6/90 = 1/15 ≈ 0.0667. Lower than 0.09 — without replacement the chance of a second blue drops from 3/10 to 2/9 after taking one blue.
1.3 — Coin then die
The coin and die are independent (with replacement is automatic — they share no pool). P(H AND 4) = 1/2 × 1/6 = 1/12 ≈ 0.083.
1.4 — At least one heart in 2 cards
P(no hearts in 2 draws) = 39/52 × 38/51 = 1482/2652 = 19/34. P(at least one heart) = 1 − 19/34 = 15/34 ≈ 0.441.
1.5 — Team, one boy AND one girl
P(B then G) = 6/10 × 4/9 = 24/90. P(G then B) = 4/10 × 6/9 = 24/90. Sum = 48/90 = 8/15 ≈ 0.533.
1.6 — 4R, 6B without replacement
(i) P(R then B) = 4/10 × 6/9 = 24/90 = 4/15.
(ii) P(B then R) = 6/10 × 4/9 = 24/90 = 4/15.
(iii) The two factors are simply rearranged (6 × 4 = 4 × 6) and the denominators are the same — multiplication is commutative, so the two ordered events have the same probability.
2 — Find the mistake
(a) The mistake is on Line 3.
(b) The problem says without replacement — so after one red is taken, both the numerator (reds left = 4) AND the denominator (total left = 7) must drop by 1.
(c) Correct second-draw probability: P(R on draw 2 | R first) = 4/7. Correct answer: P(RR) = 5/8 × 4/7 = 20/56 = 5/14 ≈ 0.357.
3 — Design your own (sample answer)
Setup: A jar has 8 vanilla and 4 chocolate biscuits.
Question: Two biscuits are taken in succession. Find P(both chocolate).
With replacement: P(CC) = 4/12 × 4/12 = 16/144 = 1/9 ≈ 0.111.
Without replacement: P(CC) = 4/12 × 3/11 = 12/132 = 1/11 ≈ 0.0909.
Compare: The without-replacement probability is smaller because after taking one chocolate, only 3 of the 11 remaining biscuits are chocolate (instead of 4 of 12).
Marking: 1 mark each for setup, with-replacement working, without-replacement working, and the comparison sentence.