Math lesson plan · 11th grade (Year 12 in England) · 60 minutes
Reverse conditional probability with a tree (tests and filters)
Statistics and probability
CCSS HSS.CP.A.3CCSS HSS.CP.B.6England: A-level Statistics
Objective and success criteria
Students will: Reverse conditional probability with a tree (tests and filters).
- I can answer questions on today's skill on my own, without help.
- I can explain each step and say why it works.
- I can check my answer and spot a mistake.
Before and after
Needs first: Use P(A | B) = P(A and B) ÷ P(B); Tree diagrams and Venn diagrams
Leads to: the next skill in this area.
Materials: Squared paper for charts, Class survey data, Sticky notes for a live chart, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 6 min
- Quick retrieval from earlier learning (Use P(A | B) = P(A and B) ÷ P(B); Tree diagrams and Venn diagrams). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- P(B) = 0.5 and P(A and B) = 0.25. Find P(A | B). → 0.5Prompt: P(A | B) = P(A and B) ÷ P(B).
- A bag has 4 red and 6 blue counters. Two are taken with replacement. What is the probability both are red? Give a fraction in simplest form. → 4/25Prompt: Multiply along the branches of the tree diagram.
- The probability it rains tomorrow is 0.3. If it rains, the probability the bus is late is 0.25. What is the probability that it rains and the bus is late? → 0.075Prompt: P(A and B) = P(B) × P(A | B).
Hook · 6 min
- Take a quick class survey (favorite fruit, travel to school) and ask what the results tell us.
- Share the objective: Reverse conditional probability with a tree (tests and filters). Read the success criteria together.
I do: model it · 12 min
- Work through the example out loud, saying why you do each step, not just what you do.
- Example: 10% of people have a condition. A test is positive for 95% of people who have it and for 10% of people who don't. Someone tests positive. What is the probability they have the condition? Give 3 decimal places.
- Step 1: P(+) = 0.1 × 0.95 + 0.9 × 0.1 = 0.185
- Step 2: 0.095 ÷ 0.185 ≈ 0.514
- Answer: 0.514
- Pause after each step and ask a student to say what comes next.
We do: guided practice · 12 min
- Do these together. Students try each step on whiteboards before you reveal it.
- Use the hints as prompts rather than giving the answer.
- 10% of people have a condition. A test is positive for 90% of people who have it and for 2% of people who don't. Someone tests positive. What is the probability they have the condition? Give 3 decimal places. → 0.833Prompt: Draw a tree: condition / no condition, then positive / negative.
- 20% of emails are spam. A filter flags 80% of spam emails and 10% of genuine emails. An email is flagged. What is the probability it is spam? Give 3 decimal places. → 0.667Prompt: Find the total probability of being flagged (two branches).
You do: independent practice · 17 min
- Students work on the student sheet. Circulate and check the first two answers of every student early.
- Students who finish: write a question of their own for a partner, with the answer.
The 8 questions are on the student sheet below, with answers in the key.
Exit ticket · 7 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- 20% of emails are spam. A filter flags 90% of spam emails and 5% of genuine emails. An email is flagged. What is the probability it is spam? Give 3 decimal places. → 0.818Prompt: Find the total probability of being flagged (two branches).
- 5% of people have a condition. A test is positive for 99% of people who have it and for 5% of people who don't. Someone tests positive. What is the probability they have the condition? Give 3 decimal places. → 0.51Prompt: Draw a tree: condition / no condition, then positive / negative.
Watch out for
- Forgetting to put data in order before finding the median.
- Reading a bar's height wrongly when the scale goes up in 2s or 5s.
Support and stretch
Support
- Review first: Use P(A | B) = P(A and B) ÷ P(B).
- Use concrete materials: squared paper for charts.
- Give the first step of each question, or a partly worked example to finish.
- Pair with a confident partner for the guided practice.
Stretch
- Explain a common mistake in writing and how to fix it.
- Write a word problem that uses today's skill, then solve a partner's.
Homework and more practice
Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-11.
Free lesson plan from eurekami.com/lesson-plan-builder