The shape of the lesson
By the end of the lesson, learners can
- The probability scale: In the lab, choose how many counters of each colour go in the bag, then pick an event.
- When it does not happen: An event either happens or it does not, so the two probabilities add up to 1.
- Reading tables and Venn diagrams: Often the outcomes are people or objects counted in a table or a Venn diagram.
- Scale: 0 impossible, 1 certain; nothing outside 0 to 1.
- Equally likely: P = number wanted \(\div\) total number.
- Not happening: \(\text{P(not A)} = 1 - \text{P(A)}\) (Extended: \(\text{P}(A')\)).
- All outcomes: the probabilities add up to 1, so a missing one is 1 minus the rest.
- Format: fraction, decimal or percentage, never "3 out of 10".
One picture every idea returns to
For equally likely outcomes, \(\text{P(event)} = \dfrac{\text{number of outcomes in the event}}{\text{total number of outcomes}}\). Probabilities lie from 0 (impossible) to 1 (certain), and \(\text{P(not A)} = 1 - \text{P(A)}\).
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, From impossible to certain.. Every probability is a number from 0 to 1. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: The probability scale | In the lab, choose how many counters of each colour go in the bag, then pick an event. Project tab 1 of the lab (Probability lab, probability scale) and let a student drive it while the class predicts. Model the worked example Counters in a bag on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: When it does not happen | An event either happens or it does not, so the two probabilities add up to 1. Model the worked example Not happening on the board, one line at a time. Your turn: The probability that it rains tomorrow is 37%. What is the probability that it does not rain? | Pairs |
| 21 to 30 min | Teach and check: Reading tables and Venn diagrams | Often the outcomes are people or objects counted in a table or a Venn diagram. Model the worked example Two-way table on the board, one line at a time. | Think-Pair-Share |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 16 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Reading tables and Venn diagrams, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Think-Pair-Share
Pose the question, give thirty seconds of silent thinking, then a minute to compare with a partner before a few pairs share with the class. The silent minute is the part that cannot be cut.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| A probability bigger than 1 or below 0 is always wrong. | If you get one, you have probably divided the wrong way round. |
| The rule \(1 - p\) only works for an event and its opposite. | P(not blue) is \(1 - \text{P(blue)}\); it is not P(red), and it is not \(1 - \text{P(red)}\). |
| In a Venn diagram, "likes coffee" includes the overlap. | Using only the "coffee only" region (10) is the usual slip. |
Introduction to probability: worksheet
6 questions, 16 marks. Written in the style of the papers; no past paper question is reproduced.
A fair six-sided dice is rolled. Find the probability of rolling (a) a 5 (b) an even number (c) a 7.
The probability that a bus is late is 0.12. Find the probability that the bus is not late.
A bag contains 5 red, 8 white and 7 black beads. One bead is taken at random. Find the probability that it is (a) white (b) not black.
A biased four-sided spinner is numbered 1 to 4. P(1) = 0.3, P(2) = 0.25, and 3 and 4 are equally likely.
(a) Find P(3). (b) Find the probability that the spinner lands on an odd number.
The table shows the pets owned by 60 people (each owns exactly one).
| Cat | Dog | |
|---|---|---|
| Adult | 12 | 18 |
| Child | 14 | 16 |
One person is chosen at random. Find the probability that the person is (a) a child (b) an adult who owns a dog (c) a cat owner.
A box contains 40 chocolates. Each is milk, dark or white. A chocolate is chosen at random: P(milk) = 0.45 and P(dark) = 0.3.
(a) How many white chocolates are in the box?
(b) 4 more dark chocolates are added to the box. Find the new probability of choosing a dark chocolate.
Answer key and marking notes
- (a) \(\tfrac{1}{6}\)
- (b) 2, 4 or 6: \(\tfrac{3}{6} = \tfrac{1}{2}\)
- (c) Impossible: 0
\(1 - 0.12 = 0.88\)
- Total \(= 20\) beads.
- (a) \(\tfrac{8}{20} = \tfrac{2}{5}\)
- (b) \(\tfrac{5 + 8}{20} = \tfrac{13}{20}\)
- (a) \(0.3 + 0.25 + 2x = 1\), so \(2x = 0.45\) and P(3) \(= 0.225\)
- (b) Odd means 1 or 3: \(0.3 + 0.225 = 0.525\)
- (a) \(14 + 16 = 30\): \(\tfrac{30}{60} = \tfrac{1}{2}\)
- (b) \(\tfrac{18}{60} = \tfrac{3}{10}\)
- (c) \(12 + 14 = 26\): \(\tfrac{26}{60} = \tfrac{13}{30}\)
- (a) P(white) \(= 1 - 0.45 - 0.3 = 0.25\), so \(0.25 \times 40 = 10\) white.
- (b) Dark at first: \(0.3 \times 40 = 12\). Now 16 dark out of 44.
- P(dark) \(= \tfrac{16}{44} = \tfrac{4}{11}\)
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
A bag contains 4 red, 7 blue and 9 white beads. One bead is taken at random. Find the probability that it is blue.
- A\(\tfrac{7}{20}\)
- B\(\tfrac{7}{13}\)
- C\(\tfrac{1}{3}\)
- D\(\tfrac{13}{20}\)
A biased spinner can land on red, blue, green or yellow. P(red) = 0.3, P(blue) = 0.25 and P(green) = 0.15. Find the probability that the spinner does NOT land on yellow.
- A0.7
- B0.3
- C0.55
- D0.45
The table shows how 55 students travel to school.
| Walk | Bus | Car | Total | |
|---|---|---|---|---|
| Year 10 | 12 | 9 | 4 | 25 |
| Year 11 | 8 | 15 | 7 | 30 |
| Total | 20 | 24 | 11 | 55 |
- A\(\tfrac{3}{11}\)
- B\(\tfrac{1}{2}\)
- C\(\tfrac{5}{8}\)
- D\(\tfrac{24}{55}\)
Thirty cards are numbered 1 to 30. One card is chosen at random. Find the probability that its number is a multiple of 4 or a multiple of 6 (or both).
- A\(\tfrac{1}{3}\)
- B\(\tfrac{2}{5}\)
- C\(\tfrac{1}{15}\)
- D\(\tfrac{7}{30}\)
In a club of 50 members, 22 play tennis (\(T\)), 18 play golf (\(G\)) and 7 play both. A member is chosen at random. Find \(\text{P}(T \cap G')\).
- A0.3
- B0.44
- C0.14
- D0.22
A bag contains only red, blue and green counters. A counter is taken at random. \(\text{P}(R) = 0.35\) and \(\text{P}(G') = 0.75\). There are 18 more blue counters than green counters. How many counters are in the bag?
- A120
- B45
- C72
- D24
Quiz key
Total \(= 4 + 7 + 9 = 20\) beads, all equally likely, and 7 are blue: \(\text{P(blue)} = \tfrac{7}{20}\). The tempting \(\tfrac{7}{13}\) compares blue with the 13 beads that are not blue, instead of with all 20. \(\tfrac{1}{3}\) treats the three colours as equally likely, and \(\tfrac{13}{20}\) is P(not blue).
P(yellow) \(= 1 - 0.3 - 0.25 - 0.15 = 0.3\), so P(not yellow) \(= 1 - 0.3 = 0.7\) (the same as \(0.3 + 0.25 + 0.15\)). The tempting 0.3 is P(yellow) itself, the answer to a different question. 0.55 adds only red and blue, and 0.45 subtracts only red and blue from 1.
15 students are in Year 11 AND travel by bus, out of 55: \(\tfrac{15}{55} = \tfrac{3}{11}\). The tempting \(\tfrac{1}{2}\) divides by the 30 Year 11 students only, and \(\tfrac{5}{8}\) divides by the 24 bus users only; the choice is from all 55. \(\tfrac{24}{55}\) counts every bus user, including Year 10.
Multiples of 4: 4, 8, 12, 16, 20, 24, 28 (7). Multiples of 6: 6, 12, 18, 24, 30 (5). 12 and 24 are in both lists, so there are \(7 + 5 - 2 = 10\) different numbers: \(\tfrac{10}{30} = \tfrac{1}{3}\). The tempting \(\tfrac{2}{5}\) counts 12 and 24 twice (\(\tfrac{12}{30}\)). \(\tfrac{1}{15}\) counts only the numbers in both lists, and \(\tfrac{7}{30}\) counts only the multiples of 4.
Tennis only \(= 22 - 7 = 15\), so \(\text{P}(T \cap G') = \tfrac{15}{50} = 0.3\). The tempting 0.44 is \(\tfrac{22}{50} = \text{P}(T)\), which includes the 7 who also play golf. 0.14 is \(\text{P}(T \cap G)\), and 0.22 is golf only, \(\text{P}(T' \cap G) = \tfrac{11}{50}\).
\(\text{P}(G) = 1 - 0.75 = 0.25\), so \(\text{P}(B) = 1 - 0.35 - 0.25 = 0.4\). The extra blue counters are \(0.4 - 0.25 = 0.15\) of the bag, so \(0.15n = 18\) and \(n = 120\). The tempting 45 treats 18 as the number of blue counters (\(18 \div 0.4\)). 72 treats it as the number of green (\(18 \div 0.25\)), and 24 divides by \(\text{P}(G') = 0.75\).
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.