126i - Problem Solving with Combined Events

Lesson 126 — Problem Solving and Consolidation: Combinations of Two Events

Strand: Probability | Descriptor: AC9M8P02 | Duration: 45 minutes

Learning Intentions

• To consolidate two-way tables, tree diagrams and Venn diagrams as tools for finding probabilities of combined events.
• To choose and justify the most efficient representation for a given practical probability problem.

Success Criteria

I can:

1. Identify structural features of a problem — sequential stages, overlapping groups, or already cross-tabulated data — that point to the best representation.
2. Construct whichever representation suits a problem and use it accurately to calculate a probability.
3. Solve a multi-step practical probability problem, justifying each stage of my method.
4. Compare two valid representation choices for the same problem and explain which is more efficient.

Warmup

(5 minutes — speed sort, pairs)

For each scenario, decide whether a two-way table, tree diagram, or Venn diagram fits best — or whether more than one would work equally well. You do not need to solve anything yet.

1. A survey asks every student whether they like maths (yes/no) and whether they like science (yes/no).
2. Two cards are drawn, one after another, from a deck, without returning the first.
3. Members of a chess club who also play soccer.
4. A coin is tossed three times in a row.
5. A council’s records cross-tabulate age group against favourite transport mode.
6. Students who study French, study Japanese, study both, or study neither.

Answers: 1. Table (already naturally cross-tabulated, two yes/no variables); 2. Tree (sequential, dependent draws); 3. Venn (overlapping group membership); 4. Tree (sequential stages); 5. Table (pre-tabulated categorical data); 6. Venn (overlapping group membership). Discuss: Q1 could also be drawn as a Venn diagram of two overlapping groups — there is not always a single correct choice.

Activities

Activity 1 — Consolidation Circuit: Four Problems, Four Choices (14 min)

Pairs. For each problem, decide on a representation, construct it, then answer.

Problem A. A vet’s clinic logged check-ups: dogs and cats. Of the dogs, needed a vaccination; of the cats, needed one. Find and .

Problem B. A pencil case has pens: blue, black. Two students each borrow one pen, one after another, without returning the first. Find and .

Problem C. A survey of students found play a sport, are in a club, and do both. Find and .

Problem D. A biased coin has . It is flipped alongside a -colour spinner: red , blue , green . Find and . (Try this one both as a table and as a tree — compare with a partner who used the other method.)

Answers:

Debrief Problem D: both a table and a two-stage tree give identical results here, since the coin and spinner act simultaneously and independently — either representation is valid whenever events are independent and unordered.

Activity 2 — Applied Task: the Science Fair (20 min)

Pairs, then whole-class share. The flagship problem of the lesson.

A regional science fair had Year 8 entrants. Of these, were in the choir, were on the debate team, and were in both.

Every student in neither group is invited to a single taster session, chosen at random with equal chance between choir and debate. Historically, of students who attend a choir taster go on to join, and of students who attend a debate taster go on to join.

(a) Represent the group membership appropriately, and state how many students are in neither group. (b) Find . (c) Harder. Estimate the expected number of “neither” students who end up joining an activity after their taster session. (d) Which representation(s) did you need for part (c), and why?

Socratic scaffolding for part (c):

Prompt	Purpose
Understand: what is being asked?	Not a probability alone — an expected count of students, built from a two-stage chance process applied to one subgroup.
What do you know?	The size of the “neither” group (from the Venn diagram), and the taster/joining probabilities (a tree of two stages).
Devise a plan	Build a tree for one “neither” student: which taster (0.5/0.5), then join or not (0.2 or 0.35). Multiply along each joining path, then add. Finally multiply by the group size.
Carry out — choir path
Carry out — debate path
Carry out — combine	per “neither” student
Carry out — scale to the group	students
Looking back — does it make sense?	sits between the two individual joining rates ( and ), as expected since it blends two equally likely paths. Rounding down to students reflects that people are whole numbers.

Answers: (a) A Venn diagram, since group membership overlaps; neither . (b) . (c) students, as scaffolded. (d) A Venn diagram was needed first, to isolate the size of the “neither” group; a tree diagram was then needed to combine the two sequential stages (which taster, then whether they join) into a single per-student probability.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. A workshop had participants: chose a morning session (of whom booked a follow-up call) and chose an afternoon session (of whom booked a follow-up call). Find .
2. A bag has yellow and purple counters. Two are drawn without replacement. Find .
3. Of diners, order dessert, order coffee, order both. Find .
4. Explain, in one or two sentences, which representation you would use for “tracking whether it rained and whether the school carnival was cancelled, recorded each day for a term”, and why.

Answers: 1. ; 2. ; 3. dessert only , coffee only , both , accounted , neither , ; 4. A two-way table — the data is already collected day by day against two fixed categorical variables (rain: yes/no; carnival: cancelled/not), not sequential stages or overlapping group membership.

Common Misconceptions

Misconception	How to pre-empt it
Believing every problem has exactly one “correct” representation.	Activity 1 Problem D and the warmup Q1 discussion both show two valid choices existing side by side.
Forgetting to subtract from the grand total when finding “neither” in a Venn-diagram problem.	Insist on writing every region (both, each “only”, and neither) before answering, as in Activity 2(a).
Applying a stage’s probability (e.g. a joining rate) to the whole population instead of the relevant subgroup only.	Always identify which group a given rate applies to before multiplying, as in Activity 2(c).
Using when and overlap.	Show the counterexample directly: this formula double-subtracts the “both” region unless is added back.
Leaving an expected-count answer as a decimal, e.g. ” students”.	Round sensibly to a whole person at the final step, and say so explicitly, as in Activity 2(c).

Enrichment — Competition-Style Problems

E1 (AMC Junior style). In a class of , study French, study Japanese, and study neither. How many study both?

Answer

E2 (Kangaroo style). A weighted coin has . It is tossed times. Find .

Answer

Three paths give exactly two heads: , , , each with probability .

E3 (Challenge). A company ships devices from two factories: Factory P makes (with a defect rate) and Factory Q makes (with a defect rate). A second inspection catches of defective devices before shipping; the rest ship anyway. Find .

Answer

Homework

1. A survey of café customers found order a hot drink, order a cold drink, order both. (a) Represent this appropriately. (b) Find . (c) Find .
2. A bag has orange and grey counters ( total). Two are drawn without replacement. Find and .
3. Two production lines: Line A makes units/day with a fault rate; Line B makes units/day with a fault rate. Find the total daily output and .
4. . If it rains, ; if not, . Find and .
5. Reasoning. Explain how you decide whether a probability problem needs a two-way table, a tree diagram, or a Venn diagram, giving one clue that points to each.
6. Reasoning. A student writes . Explain why this is only correct when and do not overlap, and state the correct version when they do.
7. Challenge. A workshop enrols people into either a coding stream ( people) or a robotics stream ( people) — the streams do not overlap. Of the coding stream, are invited to a bonus session, and of those invited attend. Of the robotics stream, are invited, and of those invited attend. Find the total expected number of attendees at bonus sessions.

Answers: 1. (a) A Venn diagram (overlapping order types) (b) hot only , (c) neither , . 2. ; . 3. Total units; faults , . 4. ; . 5. A table suits data already cross-tabulated by two categories; a tree suits events unfolding in stages, where later probabilities may depend on earlier outcomes; a Venn diagram suits overlapping group membership that all exists at once. 6. It ignores that the “both” region has been subtracted twice; the correct version is . 7. Coding: ; robotics: ; total attendees.