124i - Problem Solving with Single-Stage Events

Lesson 124 — Problem Solving with Single-Stage Events

Strand: Probability | Descriptor: AC9M7P01 | Duration: 45 minutes

Learning Intentions

• To solve multi-step problems involving probability of single-stage events.
• To design chance experiments to meet given conditions.

Success Criteria

I can:

1. Solve problems requiring several probability steps.
2. Work backwards from a probability to a sample space.
3. Design a spinner or bag to meet stated probabilities.
4. Judge whether a game is fair and adjust it.

Warmup

(6 minutes — retrieval, mini whiteboards)

A bag has red, blue and yellow counters.

1. .
2. .
3. .
4. Expected reds in draws with replacement.
5. Check that the three colour probabilities total .

Answers: 1. ; 2. ; 3. ; 4. ; 5. ✓

Note Q2 and Q3 agree. “Not blue” and “red or yellow” describe the same set of outcomes — a useful check and a hint that the complement is often the faster route.

Activities

Activity 1 — Multi-step Problems (16 min)

Pairs. Full working; probabilities as simplified fractions.

Problem 1 — The modified bag. A bag has red and blue counters.

(a) Find .

(b) Three more red counters are added. Find the new .

(c) How many red counters must be added to the original bag to make ?

Problem 2 — The spinner design. Design a spinner with equal sectors so that , , and the rest are green.

(d) How many sectors of each colour?

(e) Find .

(f) In spins, how many greens would you expect?

Problem 3 — The raffle. A raffle sells tickets. You buy .

(g) Find as a fraction and a percentage (2 d.p.).

(h) How many tickets would you need for a chance?

(i) If there are prizes, what is — approximately, treating each prize as a separate draw from the full ?

Problem 4 — The unfair game. A spinner has sectors: . Player A wins on a ; Player B wins otherwise.

(j) Find each probability.

(k) Is the game fair?

(l) Change the sector labels — not the number of sectors — to make it fair.

Socratic scaffolding for Problem 1(c):

Prompt	Purpose
What does mean about the bag?	Half the counters are red.
If we add reds, how many are red?	. And the total is .
Write the condition.	.
Solve it.	.
Check.	red of ✓
Where have you solved this shape of equation before?	Lesson 32 — variables on both sides.

(Answers: (a) ; (b) ; (c) add ; (d) red, blue, green; (e) ; (f) ; (g) ; (h) tickets; (i) about — an approximation, since the same ticket cannot win twice; (j) A: , B: ; (k) not fair; (l) relabel so exactly four sectors are s, e.g. — giving each.)

Q(i) deserves a note. The approximation is deliberate — properly, the prizes are drawn without replacement, which is a two-stage problem beyond this descriptor. Naming the approximation honestly is better than pretending it is exact.

Activity 2 — Design Tasks (14 min)

Pairs. Working backwards from probabilities to equipment.

Design each, and verify your design by calculating the probabilities.

1. A bag of counters with and .
2. A spinner with equal sectors where landing on a prime has probability .
3. A bag where , using the fewest possible counters.
4. A spinner with sectors where two players have equal chances, but one wins on exactly two different numbers.
5. A bag of counters where and there are exactly blue counters (assume only red and blue).
6. Hard: a spinner with sectors where and , with the rest “spin again”.

Socratic scaffolding for Q6:

Prompt	Purpose
How many sectors are wins?	.
How many are losses?	.
So how many are left?	— the “spin again” sector.
Check the probabilities total .	✓
Does “spin again” break the model?	It makes the game multi-stage in practice — but for a single spin, it is simply a third outcome.

(Answers: 1. red, blue, other; 2. four sectors labelled with primes among –, e.g. — which happens to be exactly the primes, giving ✓; 3. green and others — three counters; 4. e.g. with A winning on or () and B otherwise; 5. , so blue is three quarters of counters: red; 6. win, lose, spin again.)

Activity 3 — Inquiry: the Carnival Game (7 min)

Pairs, then class discussion.

A carnival stall charges 2101$152$47$ blank (nothing).

1. Find the probability of each result.
2. In plays, how much money does the stall take? How much does it pay out, on average?
3. Is the game fair to the player?
4. Change one thing to make it fair, and check your change.

Socratic scaffolding:

Prompt	Purpose
Probabilities?	Gold , silver , blank .
In plays, expected numbers of each?	gold, silver, blank.
Money in?	2 = $200$.
Money out?	230$.
So who profits?	The player — the stall loses 30100$ plays. This surprises everyone.
Q4: how would a stall fix it?	Reduce the gold prize to 12120 + 80 = $200$, exactly break-even), or cut the gold sectors, or raise the price.
Looking back	”Fair” here means expected payout equals expected income — a different sense of fair from Lesson 122’s equal probabilities.

The closing point: real carnival games are always designed so the stall’s expected payout is below its income. Working out which side a game favours is a genuine and useful application of expected frequency.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. A bag has red and blue. How many reds must be added so ?
2. Design a spinner of sectors with and . How many of each?
3. A raffle sells tickets and you buy . Find as a percentage.
4. A bag has and non-red counters. How many counters altogether?
5. Reasoning. A game costs 1P(\text{win}) = \tfrac18$1080$ plays, does the player or the operator profit?

Answers: 1. ; 2. win, bonus, other; 3. ; 4. is , so counters; 5. Income 8010$100$20$ on average, so the operator would need to change the game.

Common Misconceptions

Misconception	How to pre-empt it
Adding counters to only the numerator when modifying a bag.	Problem 1(c)‘s scaffolding — the total changes too.
Treating ” prizes” as five times the probability, exactly.	Named as an approximation in Problem 3(i).
Designing equipment without verifying the probabilities.	Verification is required in every design task.
Assuming a carnival game must favour the stall.	The inquiry’s surprise — check, don’t assume.
Confusing fairness of probability with fairness of payout.	The two senses distinguished in the inquiry’s close.
Forgetting that probabilities in a sample space total .	Used as the check in every design.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A bag has red and blue. How many reds must be added for ?

Answer

.

E2 (AMC Junior style). A spinner has sectors. and . How many sectors are neither?

Answer

Red , blue , so neither .

E3 (Challenge). A bag holds red and blue counters with . After blue counters are removed, . How many counters were there originally?

Answer

Let the original total be , with red. After removing blue: . (Check: red, blue; after removal of ✓)

E4 (Challenge). A game costs 3121$203$58120$ plays, who profits and by how much?

Answer

Income 36010 \times 20 + 30 \times 5 = 200 + 150 = $350$10120$ plays — a thin margin.

E5 (Challenge). Design a spinner of sectors and a 1$20100$ plays.

Answer

Income 100= $80k20$p100= 5k5kp = 80k = 2p = $810\times $8 = $80k = 4p = $4$.

Homework

1. A bag has red and blue. (a) . (b) After reds are added, find the new . (c) How many reds must be added to the original bag for ?
2. Design a spinner of equal sectors with , , and the rest green. State the sector counts and .
3. A raffle sells tickets; you buy . (a) as a fraction and percentage. (b) How many tickets for a chance?
4. Design a bag with using the fewest counters possible.
5. A bag has only red and blue counters, with and blue counters. How many red?
6. A spinner has sectors: gold (103$35$2200$ plays, who profits and by how much?
7. A game has , costs 2$10120$ plays, find the expected profit for the operator.
8. A spinner has sectors labelled . A wins on ; B wins otherwise. (a) Find each probability. (b) Is it fair? (c) Relabel to make it fair, keeping sectors.
9. Reasoning. Explain why adding counters to a bag changes both the numerator and the denominator of the probability.
10. Challenge. A bag holds red and blue counters with . After red counters are added, . How many counters were there originally?

Answers: Q1 — (a) (b) (c) . Q2 — red, blue, green; . Q3 — (a) (b) tickets. Q4 — green and others — five counters. Q5 — ; blue is of , so red. Q6 — income 40040 \times 10 + 60 \times 3 = 400 + 180 = $580$180$24020 \times $10 = $200$40\tfrac39 = \tfrac13\tfrac2394.59\dfrac{2n/5 + 6}{n + 6} = \dfrac12 \Rightarrow 4n + 60 = 5n + 30 \Rightarrow n = 3012301836$ ✓)