Mathematics • Year 7 • Unit 4 • Lesson 19
Two-Stage Experiments — Real World
Apply with/without replacement to real selections: a lolly bag, dealing cards, picking student leaders, a raffle barrel and selecting band members. State the procedure first, then multiply along the branches.
1. Word problems
For each problem state "with replacement" or "without replacement" before showing the multiplication line.
1.1 — Lolly bag. A bag has 6 strawberry and 4 lemon lollies. Maya takes one lolly, eats it, then takes a second. (a) Is this with or without replacement? (b) Find P(strawberry then strawberry). (c) Find P(strawberry then lemon). 4 marks
1.2 — Card deal. Two cards are dealt from a standard 52-card deck without replacement. (a) Find P(first card is a heart AND second card is a heart). (b) Find P(both red cards). 3 marks
1.3 — Student leaders. A class has 12 girls and 8 boys. The teacher writes every name on a card and picks two cards without replacement to choose the captain and vice-captain. (a) Find P(captain is a girl AND vice-captain is a girl). (b) Find P(captain is a boy AND vice-captain is a girl). 3 marks
1.4 — Raffle barrel (returned). A school raffle barrel has 50 tickets — 5 of them are "prize" tickets. A student draws one ticket, records it, then PUTS IT BACK before the next draw. (a) Is this with or without replacement? (b) Find P(2 prize tickets in 2 draws). (c) Repeat without replacement and compare. 4 marks
1.5 — Selecting band members. A school band has 7 woodwind and 5 brass players, total 12. Two players are chosen at random without replacement to perform a duet. Find P(one woodwind AND one brass, in any order). 3 marks
2. Explain your thinking
Communication matters. Use full sentences. 4 marks
2.1 A Year 7 student writes: "A bag has 3 red and 2 blue marbles. Two marbles are drawn without replacement. P(R then B) = 3/5 × 2/5 = 6/25." Using the lesson, explain (i) which factor is wrong and why, (ii) what the correct second-draw probability is, and (iii) write the right calculation and the correct answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Lolly bag (6 strawberry, 4 lemon)
(a) Without replacement — she eats the first one.
(b) P(S then S) = 6/10 × 5/9 = 30/90 = 1/3 ≈ 0.333.
(c) P(S then L) = 6/10 × 4/9 = 24/90 = 4/15 ≈ 0.267.
1.2 — Card deal
(a) P(heart then heart) = 13/52 × 12/51 = 156/2652 = 1/17 ≈ 0.0588.
(b) P(both red) = 26/52 × 25/51 = 25/102 ≈ 0.245.
1.3 — Student leaders
(a) P(G then G) = 12/20 × 11/19 = 132/380 = 33/95 ≈ 0.347.
(b) P(B then G) = 8/20 × 12/19 = 96/380 = 24/95 ≈ 0.253.
1.4 — Raffle barrel (with replacement)
(a) With replacement (the ticket is put back).
(b) P(prize then prize) = 5/50 × 5/50 = 25/2500 = 1/100 = 0.01.
(c) Without replacement: 5/50 × 4/49 = 20/2450 = 2/245 ≈ 0.00816. Slightly lower, because after pulling out a prize ticket there is one fewer prize in a smaller barrel.
1.5 — Band duet
P(W then B) = 7/12 × 5/11 = 35/132. P(B then W) = 5/12 × 7/11 = 35/132. Total = 35/132 + 35/132 = 70/132 = 35/66 ≈ 0.530.
2.1 — Explain your thinking (sample response)
(i) The second factor (2/5) is wrong. The student has kept the denominator at 5, but without replacement, after taking a red marble the pool has shrunk from 5 marbles to 4.
(ii) The correct second-draw probability is P(B | R first) = 2/4 = 1/2. There are still 2 blue marbles, but only 4 marbles remain in the bag.
(iii) Correct working: P(R then B) = 3/5 × 2/4 = 6/20 = 3/10 = 0.30.
Marking: 1 for naming the wrong factor, 1 for the correct second-draw value, 1 for the right answer, 1 for clear sentences.