Probability · Learner pages

Lesson 1 · Count equally likely outcomes

Use favourable outcomes divided by all equally likely outcomes.

Name or initials: ____________   Date: ____________

A sample space lists the outcomes of a random experiment.

1. Define the experiment. Roll a fair six-faced die once. Outcomes are 1, 2, 3, 4, 5, 6 and are equally likely.
2. Define the event. The event “greater than 4” contains outcomes 5 and 6.
3. Count both sets. There are 2 favourable outcomes among 6 equally likely outcomes, so P = 2/6 = 1/3.
4. Check the assumption. If the die is biased, simply counting faces is not enough to establish this probability. Equal likelihood is essential.

Worked example: A fair die event “multiple of 3” is {3, 6}. Its probability is 2/6 = 1/3. The event “at most 3” has three outcomes and probability 3/6 = 1/2.

1. A fair spinner has 6 equal sectors, of which 1 are blue. What is P(blue)?

   1. 1/6 · 2. 5/6 · 3. 1/7

2. A fair spinner has 8 equal sectors, of which 3 are blue. What is P(blue)?

   1. 5/8 · 2. 1/3 · 3. 3/8

3. A fair spinner has 10 equal sectors, of which 4 are blue. What is P(blue)?

   1. 4/11 · 2. 2/5 · 3. 3/5

Discuss: List an event on a fair die with probability 1/2, then create a different event with the same probability.

Lesson 2 · Use complements and bounds

Find “not” probabilities and distinguish impossible from certain.

Name or initials: ____________   Date: ____________

Every outcome either belongs to an event E or to its complement, but not both.

1. Partition all outcomes. A bag contains 3 blue and 5 red otherwise identical balls. A uniform random selection picks one ball.
2. Find the event probability. P(blue) = 3/8. The complementary event is not blue, which here means red.
3. Subtract from one. P(not blue) = 1 − 3/8 = 5/8. The two probabilities sum to 1.
4. Check endpoints. An impossible event has probability 0; a certain event has probability 1. A negative probability or one greater than 1 signals an error.

Worked example: For a fair die, P(not getting 1) = 1 − 1/6 = 5/6. P(getting a number above 6) = 0, while P(getting a number at most 6) = 1.

1. A bag has 7 otherwise identical balls, 2 blue and the rest red. One is selected uniformly at random. What is P(not blue)?

   1. 5/7 · 2. 2/7 · 3. 5/8

2. A bag has 8 otherwise identical balls, 3 blue and the rest red. One is selected uniformly at random. What is P(not blue)?

   1. 3/8 · 2. 5/9 · 3. 5/8

3. A bag has 9 otherwise identical balls, 4 blue and the rest red. One is selected uniformly at random. What is P(not blue)?

   1. 1/2 · 2. 5/9 · 3. 4/9

Discuss: Make a three-colour bag model. Explain why not red includes both of the other colours, then check complementary probabilities add to 1.

Lesson 3 · Organise a sample space without losing outcomes

Use a table or tree for a small combined experiment.

Name or initials: ____________   Date: ____________

Ordered outcomes distinguish which result belongs to which action.

1. Define independent fair actions. Toss a fair coin and spin an independent fair three-sector spinner numbered 1, 2, 3.
2. List the six ordered outcomes. They are H1, H2, H3, T1, T2, T3. Independence and fairness make these six outcomes equally likely.
3. Select the event precisely. For “heads and an even number”, only H2 qualifies. Its probability is 1/6.
4. Compare a different event. For “heads or an even number”, H1, H2, H3 and T2 qualify, giving 4/6 = 2/3. H2 must be counted only once.

Worked example: With two independent fair coins, the ordered outcomes HH, HT, TH, TT are equally likely. Exactly one head occurs in HT and TH, so its probability is 2/4 = 1/2, not 1/3.

1. A fair coin and an independent fair 2-sector spinner numbered 1 to 2 are used. What is P(heads and spinner number at most 1)?

   1. 1/4 · 2. 1/2 · 3. 3/4

2. A fair coin and an independent fair 3-sector spinner numbered 1 to 3 are used. What is P(heads and spinner number at most 1)?

   1. 1/3 · 2. 2/3 · 3. 1/6

3. A fair coin and an independent fair 4-sector spinner numbered 1 to 4 are used. What is P(heads and spinner number at most 1)?

   1. 5/8 · 2. 1/8 · 3. 1/4

Discuss: Make a 2-by-4 table for a coin and a four-sector spinner. Shade one “and” event and one “or” event, explaining any overlap.

Final check · Set 1

Try one set after the lessons. Use the other set after reviewing any errors.

1. A fair spinner has 9 equal sectors, of which 2 are blue. What is P(blue)?

   7/9 / 1/5 / 2/9

2. A fair spinner has 15 equal sectors, of which 6 are blue. What is P(blue)?

   3/8 / 2/5 / 3/5

3. A bag has 13 otherwise identical balls, 6 blue and the rest red. One is selected uniformly at random. What is P(not blue)?

   6/13 / 1/2 / 7/13

4. A bag has 9 otherwise identical balls, 2 blue and the rest red. One is selected uniformly at random. What is P(not blue)?

   7/10 / 7/9 / 2/9

5. A fair coin and an independent fair 6-sector spinner numbered 1 to 6 are used. What is P(heads and spinner number at most 1)?

   1/6 / 7/12 / 1/12

6. A fair coin and an independent fair 7-sector spinner numbered 1 to 7 are used. What is P(heads and spinner number at most 2)?

   9/14 / 1/7 / 2/7

Final check · Set 2

Try one set after the lessons. Use the other set after reviewing any errors.

1. A fair spinner has 20 equal sectors, of which 7 are blue. What is P(blue)?

   7/20 / 13/20 / 1/3

2. A fair spinner has 16 equal sectors, of which 5 are blue. What is P(blue)?

   11/16 / 5/17 / 5/16

3. A bag has 15 otherwise identical balls, 7 blue and the rest red. One is selected uniformly at random. What is P(not blue)?

   8/15 / 7/15 / 1/2

4. A bag has 11 otherwise identical balls, 3 blue and the rest red. One is selected uniformly at random. What is P(not blue)?

   3/11 / 2/3 / 8/11

5. A fair coin and an independent fair 8-sector spinner numbered 1 to 8 are used. What is P(heads and spinner number at most 3)?

   3/16 / 3/8 / 11/16

6. A fair coin and an independent fair 9-sector spinner numbered 1 to 9 are used. What is P(heads and spinner number at most 2)?

   2/9 / 11/18 / 1/9