122i - Assigning Probabilities

Lesson 122 — Assigning Probabilities to Outcomes

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

Learning Intentions

• To assign probabilities to the outcomes of a single-stage event.
• To express probabilities as fractions, decimals and percentages.

Success Criteria

I can:

1. Calculate the probability of an event with equally likely outcomes.
2. Express a probability as a fraction, decimal and percentage.
3. Place a probability on a -to- scale.
4. Use the fact that all probabilities in a sample space total .

Warmup

(6 minutes — the probability scale, projected)

Draw a line from to , marked at , and .

Place each event on it, and label:

1. The sun rises tomorrow.
2. A tossed coin lands heads.
3. Rolling a on a standard die.
4. Rolling an even number on a standard die.
5. Rolling a number less than .

Answers: 1. At — certain; 2. At ; 3. At — impossible; 4. At ; 5. At , close to but not certain.

The scale’s endpoints, named: probability runs from (impossible) to (certain). Nothing lies outside that range — a probability of or is not unlikely, it is meaningless.

Activities

Activity 1 — Explicit Instruction: the Probability Formula (14 min)

For equally likely outcomes:

The condition matters. This formula works only when every outcome in the sample space is equally likely — the distinction drawn in Lesson 121’s inquiry.

I do — a die roll. , so .

Always simplify — is correct but is the answer (Lesson 55’s habit).

Three forms of the same probability (Lesson 56):

All three are acceptable; fractions are usually exact and preferred.

The counters problem — where the equal-likelihood condition bites. A bag holds red, blue, green.

Wrong: “three colours, so .”

Right: the six counters are equally likely, not the three colours:

The check that catches this: the probabilities must total .

(The wrong answer would give too — so the total check alone does not catch it. The real check is: are the outcomes I am counting equally likely?)

The complement rule.

Rolling a die: . Often far quicker than counting the five favourable outcomes.

We do:

1. A spinner with equal sectors numbered –. Find , , , .
2. A bag of white and black marbles. Find , , and check they total .

(Answers: 1. , , , ; 2. , , total ✓)

Activity 2 — Probability Circuit (14 min)

Pairs. Every answer as a simplified fraction; convert to a decimal and percentage where asked.

Set A — dice and spinners.

1. A die: , , , .
2. A ten-sector spinner –: , , .

Set B — bags and cards.

3. A bag with red, blue: , , as fractions, decimals and percentages.
4. One card from a pack: , , , .

Set C — the complement.

5. . Find .
6. . Find .
7. A die: use the complement to find .

Set D — working backwards.

8. A bag of counters. . How many are red?
9. A spinner has some red sectors among equal sectors, and . How many are red?
10. A bag holds only red and blue counters, with . There are blue. How many counters altogether?

Socratic scaffolding for Set D Q10:

Prompt	Purpose
What does mean about the counters?	Two out of every five are blue.
If are blue, how many groups of “two” is that?	Four groups.
So how many groups of five?	Four — giving counters.
Check.	✓
Where have you used this reasoning before?	Lesson 71’s unitary method with ratios — identical structure.

(Answers: 1. , , , ; 2. , , ; 3. , ; 4. , , , ; 5. ; 6. ; 7. ; 8. ; 9. ; 10. .)

Activity 3 — Inquiry: is it Fair? (9 min)

Pairs.

A game: spin a spinner with sectors –. Player A wins on a prime; Player B wins otherwise.

1. List each player’s winning outcomes.
2. Find each player’s probability of winning.
3. Is the game fair? Explain what “fair” means mathematically.
4. Change one rule to make it fair.
5. Design a game on the same spinner where one player has exactly a chance.

Socratic scaffolding:

Prompt	Purpose
Primes between and ?	— careful, is not prime (Lesson 4).
So ?	. And .
Is the game fair, then?	Yes — equal probabilities. A pleasant surprise.
What does “fair” mean?	Every player has the same probability of winning.
Q5: which events have probability ?	Any two outcomes: e.g. “a multiple of ” .
Looking back	Fairness is a calculation, not an impression. A game can feel unfair and be fair, and vice versa.

Extension: if the game is fair but B also wins ties, is it still fair? (No — but with a single spin there are no ties. Real games often hide unfairness in tie-breaking rules.)

Checks for Understanding

(5 minutes — exit ticket)

1. A bag has green and yellow marbles. Find as a fraction, decimal and percentage.
2. A die is rolled. Find .
3. . Find .
4. A bag of counters has . How many are blue?
5. Reasoning. A bag holds red and blue counters. A student says because there are two colours. Explain the error and give the correct probability.

Answers: 1. ; 2. ; 3. ; 4. ; 5. The two colours are not equally likely — the ten counters are. .

Common Misconceptions

Misconception	How to pre-empt it
Counting categories instead of equally likely outcomes.	The counters demonstration; exit Q5.
Probabilities outside to .	The scale’s endpoints, stated as a hard limit.
Leaving fractions unsimplified.	Carried from Lesson 55; marked.
Counting the complement instead of using .	Set C shows the shortcut; both are correct but one is faster.
Believing is prime.	Inquiry Q1 — Lesson 4’s definition.
Judging fairness by feel.	Fairness is defined as equal probabilities and computed.

Enrichment — Competition-Style Problems

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

Answer

.

E2 (AMC Junior style). A spinner has equal sectors. . How many winning sectors are there?

Answer

.

E3 (Challenge). A bag holds only red and blue counters. and there are red counters. How many blue?

Answer

counters in total, so are blue.

E4 (Challenge). A die is rolled. Find .

Answer

Even: . Multiples of : . Combined, without double-counting : — four outcomes, so .

E5 (Challenge). In a bag of counters, and . What is the smallest possible , and what is ?

Answer

must make both and whole, so is a multiple of : smallest (six red, five blue). .

Homework

1. A die is rolled. Find as simplified fractions: (a) (b) (c) (d) (e) .
2. A bag has red, blue: find and as fractions, decimals and percentages, and check they total .
3. A spinner has equal sectors numbered –. Find: (a) (b) (c) (d) .
4. One card is drawn from a pack. Find: (a) (b) (c) (d) .
5. Use the complement: (a) , find (b) , find .
6. Work backwards: (a) a bag of counters with — how many green? (b) a spinner of sectors with — how many winning sectors? (c) a bag with and red counters — how many altogether?
7. A game: roll a die; A wins on a factor of , B wins otherwise. (a) Find each probability. (b) Is it fair? (c) Change one rule to make it fair.
8. Reasoning. Explain why a probability can never be greater than .
9. Reasoning. A bag has gold and silver coins. Explain why “two outcomes” does not mean a chance of gold.
10. Challenge. A bag holds red, blue and green counters with and . If there are counters, how many are green?

Answers: Q1 — (a) (b) (c) (d) (e) (). Q2 — ; ; total ✓. Q3 — (a) (b) () (c) (d) . Q4 — (a) (b) (c) (d) . Q5 — (a) (b) . Q6 — (a) (b) (c) . Q7 — (a) A: ; B: (b) not fair (c) e.g. A wins on a factor of excluding , giving and each. Q8 — a probability is a fraction of the whole sample space; the favourable outcomes can never exceed the total, so the fraction cannot exceed — and means certain. Q9 — the two outcomes are not equally likely; only one of the hundred coins is gold, so . Q10 — red , blue , so green .