125i - Probabilities of Combined Events in Practice

Lesson 125 — Determining Probabilities of Specific Outcomes in Practical Situations

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

Learning Intentions

• To determine probabilities of specific combined outcomes in practical, real-world situations using two-way tables, tree diagrams and Venn diagrams.
• To select and apply the representation best suited to a given practical situation.

Success Criteria

I can:

1. Identify a practical situation’s structure — cross-tabulated data, sequential events, or overlapping groups — to choose a suitable representation.
2. Extract the probability of a specific combined outcome from a constructed representation.
3. Interpret a calculated probability correctly in the context of the practical situation.
4. Justify my choice of representation for a given scenario.

Warmup

(5 minutes — matching task, pairs)

For each scenario, decide whether a two-way table, tree diagram, or Venn diagram would suit it best. You do not need to solve anything yet.

1. Two coins are tossed together.
2. Counters are drawn one after another from a bag, without replacement.
3. Students who play sport and/or a musical instrument.
4. Survey data crossing weather (rain/no rain) with commute mode (car/bus/walk).

Answers: 1. Table or tree (both work for simultaneous independent events); 2. Tree (sequential, dependent stages); 3. Venn (overlapping group membership); 4. Table (already cross-tabulated categorical data).

Activities

Activity 1 — Explicit Instruction: Two-way Tables in Quality Control (10 min)

I do: A factory has two machines. Machine A makes items/day with a defect rate. Machine B makes items/day with an defect rate.

	Defective	OK	Total
A
B
Total

Model: ”.” ”.”

We do: Together find and .

You do: A school’s commuters: bus riders total, on time, late; car riders total, on time, late. Find and .

(We do answers: ; . You do answers: ; .)

Activity 2 — Explicit Instruction: Tree Diagrams in Sequential Practical Situations (10 min)

I do: A weather forecast gives . If it rains, ; if not, (from a distant storm). Find and .

We do: A basketball player has . If she makes the first, ; if she misses, . Together find and .

You do: A quality inspection: . If it passes, ; if it fails, . Find and .

(We do answers: ; . You do answers: ; .)

Activity 3 — Applied Task: Combining Representations (16 min)

Pairs, then whole-class share.

A streaming service surveyed subscribers: subscribe to Movies, subscribe to Sports, and subscribe to both.

(a) Represent this information appropriately. State which representation you chose and why. (b) Find . (c) Find . (d) Harder. Historically, of “Sports only” subscribers accept a Movies discount when offered — but this rises to during a promotional month. Estimate the number of new dual subscribers gained in a normal month versus a promotional month. Which representations did you need to combine to answer this?

Socratic scaffolding for part (d):

Prompt	Purpose
Understand the problem	You need the count of Sports-only subscribers first, then apply an acceptance rate to that group only.
Devise a plan	Read the Sports-only count from the Venn diagram; multiply by each acceptance rate.
Carry out — normal month	new dual subscribers (people are whole numbers).
Carry out — promotional month	new dual subscribers.
Compare	The promotional month gains roughly – more dual subscribers.
Looking back — does the scaling make sense?	and — consistent, since the same base group is simply converting at a higher rate.

Answers: Venn regions: Movies only , Sports only , both , neither . (a) A Venn diagram, since membership of the two packages overlaps. (b) . (c) . (d) As scaffolded — this task needed the Venn diagram (to isolate the Sports-only group) and a rate/tree-style calculation (to apply the acceptance probability to that group).

Checks for Understanding

(6 minutes — exit ticket, collected)

1. A cafe’s data: of customers, order coffee, order a pastry, order both. Find .
2. Using Q1’s data, find .
3. . If sunny, ; if not sunny, . Find .
4. Explain which representation you would use for “a survey of which pets (dog, cat, both, neither) people own”, and why.

Answers: 1. Coffee only , ; 2. Pastry only ; neither , ; 3. ; 4. A Venn diagram — pet ownership involves overlapping group membership, not sequential stages or pre-tabulated categories.

Common Misconceptions

Misconception	How to pre-empt it
Choosing a tree diagram for overlapping (simultaneous) group membership.	Ask “does one thing happen after another, or can both just be true at once?” before choosing a tool.
Treating already-tabulated survey data as if it needed a fresh Venn diagram.	If row and column totals are already given, a two-way table is usually the more direct fit.
Applying a practical rate (e.g. a discount acceptance rate) to the whole population instead of the relevant subgroup only.	Always identify which group the given percentage applies to before multiplying, as in Activity 3(d).
Ignoring real-world rounding — leaving an answer like ” subscribers”.	Round sensibly to a whole person/item at the final step, and say so explicitly.
Forgetting to interpret the final number in context (units, what it represents).	Require a one-sentence interpretation alongside every practical calculation.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). A shipment’s parts come from two suppliers: Supplier X provides of parts with a defect rate; Supplier Y provides with a defect rate. Find the overall probability a randomly chosen part is defective.

Answer

E2 (Kangaroo style). A goalkeeper has . If she saves it, ; if not, . Find .

Answer

E3 (Challenge). In Activity 3, suppose the promotional acceptance rate of runs for separate months, with new “Sports only” subscribers appearing fresh each month. Estimate the total number of new dual subscribers gained across the promotional months.

Answer

Homework

1. A pharmacy has two pharmacists. Pharmacist A fills scripts/week with a error rate; Pharmacist B fills scripts/week with a error rate. Find the total scripts filled, and .
2. . If on time, ; if late, . Find .
3. students: play an instrument, do art club, do both. Find .
4. Using Q3’s data, find .
5. Reasoning. Explain how you decide whether a practical situation calls for a tree diagram or a Venn diagram, giving one example of each.
6. Challenge. In a vaccine trial, of participants receive the real vaccine and a placebo. Among vaccinated participants, still get sick; among placebo participants, get sick. Find the overall probability a random participant gets sick. Then find what fraction of sick participants had been vaccinated.

Answers: 1. Total scripts; errors , so . 2. . 3. Both , instrument only , art only , neither ; . 4. . 5. A tree diagram suits events that unfold in stages, where later probabilities may depend on earlier outcomes (e.g. drawing counters without replacement); a Venn diagram suits events that describe overlapping membership of groups that all exist at once (e.g. sport and music participation). 6. ; of the sick participants, the vaccinated share is , i.e. about .