The shape of the lesson
By the end of the lesson, learners can
- Sample space diagrams: In the lab, choose two dice and the rule for combining them (add, multiply, difference), then pick an event.
- Venn diagrams with two sets: In the lab, click a region of the Venn diagram to see its count and its probability.
- Set notation in probability: (Extended) Venn questions may use set notation: \(A \cap B\) is "A and B" (the overlap), \(A \cup B\) is "A or B or both" (everything inside the circles), and \(A'\) is "not A".
- Sample space: a grid of every pair of outcomes; each cell equally likely.
- Order matters: (2, 6) and (6, 2) are separate cells.
- Venn: overlap first, then each circle minus the overlap, then the outside.
- Check: all regions add up to the total.
- Notation (Extended): \(\cap\) and, \(\cup\) or, \('\) not.
One picture every idea returns to
In a sample space diagram every cell is equally likely, so \(\text{P} = \dfrac{\text{cells in the event}}{\text{total cells}}\). In a Venn diagram, fill the overlap first, then the rest of each circle, then the outside; the probability is the region's count over the total.
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, List every outcome.. When two things happen together, such as two dice or a coin and a spinner, a sample space diagram lists every equally likely outcome. 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: Sample space diagrams | In the lab, choose two dice and the rule for combining them (add, multiply, difference), then pick an event. Project tab 1 of the lab (Combined probability lab, sample space) and let a student drive it while the class predicts. Model the worked example Two dice on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Venn diagrams with two sets | In the lab, click a region of the Venn diagram to see its count and its probability. Project tab 3 of the lab (Combined probability lab, Venn diagrams) and let a student drive it while the class predicts. Model the worked example Filling a Venn diagram on the board, one line at a time. | Pairs |
| 21 to 30 min | Teach and check: Set notation in probability | (Extended) Venn questions may use set notation: \(A \cap B\) is "A and B" (the overlap), \(A \cup B\) is "A or B or both" (everything inside the circles), and \(A'\) is "not A". Model the worked example Reading the notation (Extended) on the board, one line at a time. Your turn: (Extended) \(\text{P}(A) = 0.45\), \(\text{P}(B) = 0.35\) and \(\text{P}(A \cup B) = 0.6\). Find \(\text{P}(A \cap B)\). | Rally Coach |
| 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, 17 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 Set notation in probability, 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.
Rally Coach
Pairs share one worksheet. One partner solves the first question aloud while the other coaches, checking each line and offering a prompt rather than the answer; they swap roles for the next question.
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 |
|---|---|
| (2, 6) and (6, 2) are DIFFERENT outcomes: the first dice shows 2 in one and 6 in the other. | Counting them once is why students get \(\tfrac{3}{36}\) instead of \(\tfrac{5}{36}\) for a total of 8. |
| Do not put 17 in the football-only region: 17 is the WHOLE football circle, which already includes the 5 who play both. | Return to Venn diagrams with two sets: ask a pair to state the correct version and show it on the lab before moving on. |
| \(\text{P}(A \cup B)\) is NOT \(\text{P}(A) + \text{P}(B)\) when the sets overlap: adding counts the overlap twice. | Use the regions of the diagram. |
Combined events: sample spaces and Venn diagrams: worksheet
6 questions, 17 marks. Written in the style of the papers; no past paper question is reproduced.
A fair coin is thrown and a fair spinner numbered 1, 2, 3 is spun.
(a) List all the possible outcomes. (b) Find the probability of a tail and a 3.
Two fair dice are rolled and the scores added. Find the probability that the total is (a) 7 (b) a prime number.
Of 50 students, 28 are in the maths club, 19 are in the science club and 11 are in both.
A student is chosen at random. Find the probability that the student is in (a) neither club (b) exactly one club.
In a group of 32 students, 20 swim, 15 cycle and 4 do neither.
Find the probability that a student chosen at random swims but does not cycle.
(Extended) \(\text{P}(A) = 0.5\), \(\text{P}(B) = 0.4\) and \(\text{P}(A \cap B) = 0.15\). Find (a) \(\text{P}(A \cup B)\) (b) \(\text{P}(A \cup B)'\) (c) \(\text{P}(A' \cap B)\).
In a game, two fair dice are rolled and the score is the larger of the two numbers (if they are equal, the score is that number).
(a) Find the probability that the score is 6. (b) Find the probability that the score is 3 or less. (c) The game is played 180 times. How many scores of 6 would you expect?
Answer key and marking notes
- (a) H1, H2, H3, T1, T2, T3
- (b) One of 6 equally likely outcomes: \(\tfrac{1}{6}\)
- (a) (1,6), (2,5), (3,4), (4,3), (5,2), (6,1): \(\tfrac{6}{36} = \tfrac{1}{6}\)
- (b) Totals 2, 3, 5, 7, 11 occur \(1 + 2 + 4 + 6 + 2 = 15\) times: \(\tfrac{15}{36} = \tfrac{5}{12}\)
- Maths only \(28 - 11 = 17\), science only \(19 - 11 = 8\), both 11.
- Neither \(= 50 - (17 + 11 + 8) = 14\).
- (a) \(\tfrac{14}{50} = \tfrac{7}{25}\)
- (b) \(\tfrac{17 + 8}{50} = \tfrac{25}{50} = \tfrac{1}{2}\)
- Let \(x\) do both: \((20 - x) + x + (15 - x) + 4 = 32\), so \(39 - x = 32\) and \(x = 7\).
- Swim only \(= 20 - 7 = 13\).
- Probability \(= \tfrac{13}{32}\)
- A only \(= 0.35\), B only \(= 0.25\), both 0.15.
- (a) \(0.35 + 0.15 + 0.25 = 0.75\)
- (b) \(1 - 0.75 = 0.25\)
- (c) B but not A: \(0.25\)
- (a) Score 6 when at least one dice shows 6: the last row and column of the grid, \(6 + 6 - 1 = 11\) cells, so \(\tfrac{11}{36}\).
- (b) Both dice 3 or less: \(3 \times 3 = 9\) cells, so \(\tfrac{9}{36} = \tfrac{1}{4}\).
- (c) \(\tfrac{11}{36} \times 180 = 55\)
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.
Two fair six-sided dice are rolled and the scores are added. Find the probability that the total is 9.
- A\(\tfrac{1}{9}\)
- B\(\tfrac{1}{18}\)
- C\(\tfrac{1}{11}\)
- D\(\tfrac{1}{4}\)
A fair coin is thrown and a fair spinner numbered 1 to 5 is spun. Find the probability of a tail and a number greater than 3.
- A\(\tfrac{1}{5}\)
- B\(\tfrac{2}{5}\)
- C\(\tfrac{9}{10}\)
- D\(\tfrac{2}{7}\)
In a class of 32 students, 19 like maths, 14 like science and 5 like neither. A student is chosen at random. Find the probability that the student likes exactly one of the two subjects.
- A\(\tfrac{21}{32}\)
- B\(\tfrac{3}{16}\)
- C\(\tfrac{27}{32}\)
- D\(\tfrac{5}{32}\)
The Venn diagram shows the numbers of people in sets A and B. One person is chosen at random. Find the probability that the person is in set B.
- A\(\tfrac{11}{25}\)
- B\(\tfrac{7}{25}\)
- C\(\tfrac{4}{5}\)
- D\(\tfrac{11}{20}\)
\(\text{P}(A) = 0.55\), \(\text{P}(B) = 0.4\) and \(\text{P}(A \cup B)' = 0.2\). Find \(\text{P}(A' \cap B)\).
- A0.25
- B0.15
- C0.4
- D0.45
Two fair spinners are each numbered 1, 2, 3, 4. Both are spun and the two numbers are added. Find the probability that the total is a prime number.
- A\(\tfrac{9}{16}\)
- B\(\tfrac{4}{7}\)
- C\(\tfrac{1}{2}\)
- D\(\tfrac{7}{16}\)
Quiz key
The cells with total 9 are (3, 6), (4, 5), (5, 4) and (6, 3): 4 cells out of 36, so \(\tfrac{4}{36} = \tfrac{1}{9}\). The tempting \(\tfrac{1}{18}\) counts each pair in only one order, but (3, 6) and (6, 3) are different outcomes. \(\tfrac{1}{11}\) treats the 11 possible totals as equally likely, and \(\tfrac{1}{4}\) uses 9 cells instead of 4.
There are \(2 \times 5 = 10\) equally likely outcomes. Tail and more than 3: T4 and T5, so \(\tfrac{2}{10} = \tfrac{1}{5}\). The tempting \(\tfrac{2}{5}\) looks only at the spinner and ignores the coin. \(\tfrac{9}{10}\) adds \(\tfrac{1}{2} + \tfrac{2}{5}\), and \(\tfrac{2}{7}\) counts the 7 symbols H, T, 1, 2, 3, 4, 5 as outcomes.
\((19 - x) + x + (14 - x) + 5 = 32\) gives \(38 - x = 32\), so \(x = 6\). Maths only \(= 13\), science only \(= 8\), so exactly one \(= \tfrac{13 + 8}{32} = \tfrac{21}{32}\). The tempting \(\tfrac{3}{16}\) is the overlap \(\tfrac{6}{32}\) (both subjects). \(\tfrac{27}{32}\) is at least one subject, which includes the 6 who like both, and \(\tfrac{5}{32}\) is neither.
In B: \(4 + 7 = 11\) people. Total \(= 9 + 4 + 7 + 5 = 25\). So \(\text{P}(B) = \tfrac{11}{25}\). The tempting \(\tfrac{7}{25}\) uses only the B-only region and forgets the overlap. \(\tfrac{4}{5}\) counts everyone inside either circle, and \(\tfrac{11}{20}\) leaves the 5 people outside the circles out of the total.
\(\text{P}(A \cup B) = 1 - 0.2 = 0.8\). Adding \(0.55 + 0.4 = 0.95\) counts the overlap twice, so \(\text{P}(A \cap B) = 0.95 - 0.8 = 0.15\). Then B only \(= 0.4 - 0.15 = 0.25\). The tempting 0.15 is the overlap \(\text{P}(A \cap B)\) itself. 0.4 is \(\text{P}(B)\), which still includes the overlap, and 0.45 is \(\text{P}(A')\).
The prime totals are 2, 3, 5 and 7. In the 16 cells: total 2 once, 3 twice, 5 four times and 7 twice, which is 9 cells, so \(\tfrac{9}{16}\). The tempting \(\tfrac{4}{7}\) treats the 7 possible totals as equally likely. \(\tfrac{1}{2}\) forgets that 2 is prime, and \(\tfrac{7}{16}\) is the probability that the total is NOT prime.
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.