The shape of the lesson
By the end of the lesson, learners can
- Drawing the tree: In the lab, set up a bag of counters and build the tree one pick at a time.
- Independent events: Trees work for any events that follow each other, not just counters.
- Without replacement: (Extended) If the first item is NOT put back, the second pick has one fewer item in the bag, and one fewer of the colour already taken.
- Draw: probabilities beside the branches, outcomes at the ends.
- Check: each set of branches adds to 1.
- And: multiply along a path.
- Or: add the paths that give the event (both orders!).
- At least one: 1 minus P(none).
- Without replacement (Extended): reduce the totals on the second branches.
One picture every idea returns to
Probabilities go beside the branches and outcomes at the ends. Multiply ALONG a path for 'this and then that'; ADD the results of different paths for 'this or that'. The branches from any one point add up to 1.
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, Multiply along, add down.. A tree diagram shows one event after another. 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: Drawing the tree | In the lab, set up a bag of counters and build the tree one pick at a time. Project tab 2 of the lab (Combined probability lab, tree diagrams) and let a student drive it while the class predicts. Model the worked example With replacement on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Independent events | Trees work for any events that follow each other, not just counters. Model the worked example Two tests on the board, one line at a time. Your turn: The probability that it rains on any day is 0.3. Find the probability that it rains on exactly one of two days. | Pairs |
| 21 to 30 min | Teach and check: Without replacement | (Extended) If the first item is NOT put back, the second pick has one fewer item in the bag, and one fewer of the colour already taken. Model the worked example Two green (Extended) on the board, one line at a time. Your turn: (Extended) For the same bag, find the probability that at least one sweet is red. | Numbered Heads Together |
| 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, 18 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 Without replacement, 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.
Numbered Heads Together
Teams of four number off. Put up a question, give the team time to make sure every member can answer, then call a number: that student answers for the team, so nobody can leave the thinking to someone else.
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 |
|---|---|
| Forgetting one of the two orders. 'One red and one blue' can happen as RB OR as BR, so there are two paths to add. | Return to Drawing the tree: ask a pair to state the correct version and show it on the lab before moving on. |
| Adding along a path instead of multiplying gives nonsense: \(0.8 + 0.6 = 1.4\) is bigger than 1. | Along a path, always multiply. |
| Without replacement, BOTH the numerator and the denominator can change. | After a green is taken, green becomes \(\tfrac{2}{7}\) but red becomes \(\tfrac{5}{7}\), not \(\tfrac{5}{8}\). |
Tree diagrams: worksheet
6 questions, 18 marks. Written in the style of the papers; no past paper question is reproduced.
A fair coin is thrown twice. Find the probability of (a) two heads (b) one head and one tail.
The probability that Ali wins a game of chess is 0.65. He plays two games; the results are independent.
Find the probability that he wins (a) both games (b) neither game (c) at least one game.
A bag holds 2 white and 3 black balls. A ball is taken, replaced, and a second ball is taken.
Find the probability that the two balls are different colours.
(Extended) A box contains 6 milk and 4 dark chocolates. Two are taken at random without replacement.
Find the probability that (a) both are dark (b) at least one is milk.
On her way to work, Zara passes two sets of traffic lights. The probability that the first is red is 0.4 and that the second is red is 0.7, independently.
(a) Find the probability that she is stopped by at least one red light. (b) In 20 journeys, how many times would she expect to meet no red light?
(Extended) A bag contains \(x\) red counters and 4 blue counters. Two counters are taken at random without replacement. The probability that both are blue is \(\tfrac{2}{15}\).
Show that \(x^2 + 7x - 78 = 0\), and find the number of red counters.
Answer key and marking notes
- (a) \(\tfrac{1}{2} \times \tfrac{1}{2} = \tfrac{1}{4}\)
- (b) HT or TH: \(\tfrac{1}{4} + \tfrac{1}{4} = \tfrac{1}{2}\)
- (a) \(0.65 \times 0.65 = 0.4225\)
- (b) \(0.35 \times 0.35 = 0.1225\)
- (c) \(1 - 0.1225 = 0.8775\)
- WB: \(\tfrac{2}{5} \times \tfrac{3}{5} = \tfrac{6}{25}\); BW: \(\tfrac{3}{5} \times \tfrac{2}{5} = \tfrac{6}{25}\).
- Total \(\tfrac{12}{25}\)
- (a) \(\tfrac{4}{10} \times \tfrac{3}{9} = \tfrac{12}{90} = \tfrac{2}{15}\)
- (b) The only other outcome is both dark: \(1 - \tfrac{2}{15} = \tfrac{13}{15}\)
- P(no red) \(= 0.6 \times 0.3 = 0.18\).
- (a) \(1 - 0.18 = 0.82\)
- (b) \(0.18 \times 20 = 3.6\) journeys
- \(\dfrac{4}{x + 4} \times \dfrac{3}{x + 3} = \dfrac{2}{15}\)
- \(12 \times 15 = 2(x + 4)(x + 3)\), so \(90 = x^2 + 7x + 12\).
- \(x^2 + 7x - 78 = 0\)
- \((x + 13)(x - 6) = 0\), so \(x = 6\) (a number of counters cannot be negative).
- There are 6 red counters.
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 holds 4 green and 6 yellow counters. A counter is taken at random, its colour is noted and it is put back. A second counter is then taken. Find the probability that both counters are green.
- A\(\tfrac{4}{25}\)
- B\(\tfrac{2}{15}\)
- C\(\tfrac{4}{5}\)
- D\(\tfrac{12}{25}\)
The tree diagram shows the probabilities that Ravi's alarm rings and, independently, that his bus is on time. Find the probability that exactly one of these two things happens (the alarm rings or the bus is on time, but not both).
- A0.3
- B0.225
- C0.675
- D0.325
The probability that it rains on any day is 0.35, independently of other days. Find the probability that it rains on at least one of two days.
- A0.5775
- B0.7
- C0.1225
- D0.455
A fair coin is thrown three times. Find the probability of exactly one tail.
- A\(\tfrac{3}{8}\)
- B\(\tfrac{1}{8}\)
- C\(\tfrac{1}{4}\)
- D\(\tfrac{7}{8}\)
A bag holds 6 red and 4 blue marbles. Two marbles are taken at random without replacement. Find the probability that they are different colours.
- A\(\tfrac{8}{15}\)
- B\(\tfrac{4}{15}\)
- C\(\tfrac{12}{25}\)
- D\(\tfrac{7}{15}\)
A box contains 8 light bulbs, 3 of which are faulty. Two bulbs are taken at random without replacement. Find the probability that at least one of them is faulty.
- A\(\tfrac{9}{14}\)
- B\(\tfrac{39}{64}\)
- C\(\tfrac{15}{28}\)
- D\(\tfrac{3}{28}\)
Quiz key
\(\text{P(G)} = \tfrac{4}{10} = \tfrac{2}{5}\) on both picks, so \(\text{P(GG)} = \tfrac{2}{5} \times \tfrac{2}{5} = \tfrac{4}{25}\). The tempting \(\tfrac{2}{15}\) is \(\tfrac{4}{10} \times \tfrac{3}{9}\), which would be right only WITHOUT replacement. \(\tfrac{4}{5}\) adds along the path instead of multiplying, and \(\tfrac{12}{25}\) is the probability of one of each colour.
Rings then late: \(0.9 \times 0.25 = 0.225\). Silent then on time: \(0.1 \times 0.75 = 0.075\). Add: \(0.225 + 0.075 = 0.3\). The tempting 0.225 is only one of the two paths. 0.675 is both happening (\(0.9 \times 0.75\)), and 0.325 is "not both", \(1 - 0.675\), which also includes the path where neither happens.
P(no rain on a day) \(= 1 - 0.35 = 0.65\), so P(dry both days) \(= 0.65 \times 0.65 = 0.4225\). P(at least one) \(= 1 - 0.4225 = 0.5775\). The tempting 0.7 adds \(0.35 + 0.35\), which double counts the days when it rains twice. 0.1225 is rain on both days, and 0.455 is rain on exactly one day.
Each path has probability \(\tfrac{1}{2} \times \tfrac{1}{2} \times \tfrac{1}{2} = \tfrac{1}{8}\). Exactly one tail: THH, HTH, HHT, which is 3 paths, so \(\tfrac{3}{8}\). The tempting \(\tfrac{1}{8}\) counts only one of the three paths. \(\tfrac{1}{4}\) treats 0, 1, 2 and 3 tails as equally likely, and \(\tfrac{7}{8}\) is at least one tail.
RB: \(\tfrac{6}{10} \times \tfrac{4}{9} = \tfrac{24}{90}\). BR: \(\tfrac{4}{10} \times \tfrac{6}{9} = \tfrac{24}{90}\). Total \(= \tfrac{48}{90} = \tfrac{8}{15}\). The tempting \(\tfrac{4}{15}\) uses only one of the two orders. \(\tfrac{12}{25}\) treats the picks as WITH replacement, and \(\tfrac{7}{15}\) is the probability of the same colour.
P(both good) \(= \tfrac{5}{8} \times \tfrac{4}{7} = \tfrac{20}{56} = \tfrac{5}{14}\), so P(at least one faulty) \(= 1 - \tfrac{5}{14} = \tfrac{9}{14}\). The tempting \(\tfrac{39}{64}\) uses \(1 - \tfrac{5}{8} \times \tfrac{5}{8}\), as if the first bulb were put back. \(\tfrac{15}{28}\) is exactly one faulty, missing the both-faulty path, and \(\tfrac{3}{28}\) is both faulty.
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.