125i - Consolidation Sample Space and Probability

Lesson 125 — Consolidation and Check: Sample Space and Probability

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

Learning Intentions

• To consolidate sample spaces, probability calculation and expected frequency.
• To apply probability reasoning to unfamiliar situations.

Success Criteria

I can:

1. List a sample space and identify an event’s outcomes.
2. Calculate probabilities and express them in three forms.
3. Predict expected frequencies and interpret relative frequencies.
4. Design and analyse chance situations.

Warmup

(6 minutes — rapid fire, mini whiteboards)

1. for rolling a twelve-sided die.
2. on that die.
3. A bag of red, blue: .
4. Expected s when a die is rolled times.
5. A coin tossed times gives heads. Relative frequency?

Answers: 1. ; 2. ; 3. ; 4. ; 5. .

Activities

Activity 1 — Mixed Skills Circuit (16 min)

Stations spanning Lessons 121–124.

Station A — Sample spaces and events.

1. List the sample space for drawing a letter from PROBABILITY. State .
2. For a die –, list the outcomes in “a factor of ” and in “not a multiple of “.
3. Give an example of a certain event and an impossible event for a spinner numbered –.

Station B — Calculate.

4. A bag of red, blue, green: find , , , as fractions.
5. One card from a pack: , , .
6. A spinner of sectors –: , , expressed as a fraction, decimal and percentage.

Station C — Expected and relative frequency.

7. A die rolled times: expected s? Expected odd results?
8. A coin tossed times gives heads. Relative frequency (3 d.p.)? Compare with theory.
9. A pin dropped times lands point-up times. Estimate .

Station D — Design and work backwards.

10. A bag of counters with : how many red?
11. A spinner of sectors with and : sector counts and ?
12. A bag has red and blue. How many reds must be added for ?

Socratic scaffolding for Station D Q12:

Prompt	Purpose
After adding reds, how many are red? How many altogether?	and .
Write the condition.	.
Solve.	.
Check.	red of ✓

(Answers: 1. A,B,I,L,O,P,R,T,Y, ; 2. and ; 3. certain e.g. “less than ”; impossible e.g. “greater than ”; 4. , , ; 5. , , ; 6. ; ; 7. ; ; 8. — a little above , unremarkable; 9. ; 10. ; 11. win, bonus, neither, ; 12. .)

Activity 2 — Extended Problems (14 min)

Pairs. Multi-step; full working.

Problem 1 — The class survey. In a class of , students walk to school, catch the bus, and the rest are driven. One student is chosen at random.

(a) Find , , , and check they total .

(b) Find two ways.

(c) If a student were chosen at random on each of school days, how many times would you expect a bus-catcher?

Problem 2 — The prize wheel. A wheel has equal sectors: jackpot, major prizes, minor prizes, blanks.

(d) Find each probability.

(e) Find .

(f) In spins, how many of each result would you expect?

(g) The jackpot pays 50$10$2$480$ spins, who profits?

Problem 3 — Testing a claim. A manufacturer claims its dice are fair. A student rolls one times and records sixes.

(h) How many sixes were expected?

(i) Find the relative frequency of sixes.

(j) Is this strong evidence the die is loaded? Explain.

(k) What would you do next?

Socratic scaffolding for Problem 3:

Prompt	Purpose
Expected sixes?	.
Observed relative frequency?	against .
Is far from ?	Noticeably above, but a single run varies.
Compare with Lesson 123’s coin discussion.	against is suspicious but not conclusive from one experiment.
What next?	Roll many more times — the reliable way to distinguish luck from bias (Lesson 120’s principle).
Could the die still be fair?	Yes. Unusual results happen to fair dice; that is exactly what “chance” means.

(Answers: (a) , , ; total ✓; (b) , or ✓; (c) ; (d) ; (e) ; (f) jackpot, major, minor, blank; (g) income 3205(50) + 15(10) + 25(2) = 250 + 150 + 50 = $450$130$; (h)–(k) as scaffolded.)

Activity 3 — Synthesis (5 min)

Whole class, closing the P01 block.

Complete from memory:

Question	Answer
What are the two ends of the probability scale?
What must all probabilities in a sample space total?
?
Expected frequency ?
Relative frequency ?
Does two outcomes mean a fifty-fifty chance?

Completed: (impossible) and (certain); ; ; number of trials; occurrences trials; no — only if the outcomes are equally likely.

Checks for Understanding

(7 minutes — exit ticket, collected)

1. A bag has red, blue, green. Find and .
2. A die is rolled times. Expected number of results less than ?
3. A spinner is spun times, landing on red times. Estimate .
4. A bag of counters has . How many yellow?
5. A bag has red and blue. How many reds must be added for ?
6. Reasoning. A coin tossed times gives heads. Is the coin biased? Explain what you would need to decide.

Answers: 1. and ; 2. , so ; 3. ; 4. ; 5. ; 6. Not on this evidence — heads in is above the expected but well within ordinary variation for a fair coin. Many more tosses would be needed; a persistent deviation across hundreds of tosses would be convincing.

Common Misconceptions

Misconception	How to pre-empt it
Two outcomes means fifty-fifty.	The synthesis table’s final row.
Probabilities outside –.	The scale’s endpoints, recalled from memory.
Adding to the numerator only when modifying a bag.	Station D Q12 and its scaffolding.
Treating one unusual result as proof of bias.	Problem 3 and exit Q6.
Forgetting to check that probabilities total .	Built into Problem 1(a).
Assuming a prize game favours the operator.	Problem 2(g) — check, don’t assume.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A bag has red, blue and green. Find .

Answer

.

E2 (AMC Junior style). A die is rolled times. How many results would you expect to be prime?

Answer

Primes on a die: — so , giving .

E3 (Challenge). A bag holds only red and blue counters. Adding red counters changes from to . How many counters were there originally?

Answer

Let the original total be , with red. Then . (Check: of , then of ✓)

E4 (Challenge). A wheel has sectors: pay 256$5$3240$ spins.

Answer

Income 72020 \times 25 + 60 \times 5 = 500 + 300 = $800$80$ — the game favours the player and would need redesigning.

E5 (Challenge). Explain why a fair die producing sixes in rolls is possible but unusual, and what would make you confident it was loaded.

Answer

Twenty sixes are expected; is a substantial excess but any result is possible with a fair die. Confidence would come from repeating the experiment: if rolls gave around sixes (a relative frequency near rather than ), chance alone would be an implausible explanation. More trials separate luck from bias.

Homework

1. List the sample space and : (a) drawing a letter from CALCULATOR (b) rolling a -sided die (c) choosing a season at random.
2. A bag has red, blue, yellow. Find: (a) (b) (c) (d) check the three colour probabilities total .
3. One card from a pack: (a) (b) (c) .
4. A spinner of sectors –: (a) (b) (c) express (a) as a decimal and percentage.
5. Expected frequency: (a) a die rolled times — how many s? (b) a coin tossed times — how many tails? (c) a bag of red and blue drawn times with replacement — how many reds?
6. A spinner spun times lands on blue times. (a) Relative frequency (3 d.p.). (b) If the spinner has equal sectors, how many are probably blue?
7. A bag of counters has . (a) How many green? (b) How many greens must be added for ?
8. A wheel of sectors: pays 304$5$2200$ spins, who profits and by how much?
9. Reasoning. Explain why a relative frequency from trials is more trustworthy than one from .
10. Reasoning. Explain why “there are three colours in the bag” does not mean each has probability .
11. Challenge. A bag holds red and blue counters with . After blue counters are removed, . How many counters were there originally?

Answers: Q1 — (a) A,C,L,O,R,T,U, (b) (c) . Q2 — (a) (b) (c) (d) ✓. Q3 — (a) (b) (c) . Q4 — (a) (b) () (c) . Q5 — (a) (b) (c) . Q6 — (a) (b) about of (). Q7 — (a) (b) . Q8 — income 40010 \times 30 + 40 \times 5 = 300 + 200 = $500$10020n\tfrac{n}{3}\dfrac{n/3}{n-8} = \dfrac12 \Rightarrow 2n/3 = n - 8 \Rightarrow n/3 = 8 \Rightarrow n = 24816816$ ✓)