• 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.

1. P(B) = 0.4 and P(A and B) = 0.1. Find P(A | B). → 0.25Prompt: P(A | B) = P(A and B) ÷ P(B).
2. A bag has 7 red and 7 blue counters. Two are taken with replacement. What is the probability both are red? Give a fraction in simplest form. → 1/4Prompt: Multiply along the branches of the tree diagram.
3. The probability it rains tomorrow is 0.4. If it rains, the probability the bus is late is 0.2. What is the probability that it rains and the bus is late? → 0.08Prompt: P(A and B) = P(B) × P(A | B).

• 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.

• 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.

• Do these together. Students try each step on whiteboards before you reveal it.
• Use the hints as prompts rather than giving the answer.

1. 5% of people have a condition. A test is positive for 90% 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. → 0.486Prompt: Draw a tree: condition / no condition, then positive / negative.
2. 30% 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. → 0.885Prompt: Find the total probability of being flagged (two branches).

• 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 6 questions are on the student sheet below, with answers in the key.

• Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.

1. 1% of people have a condition. A test is positive for 95% 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. → 0.161Prompt: Draw a tree: condition / no condition, then positive / negative.
2. 30% of emails are spam. A filter flags 80% of spam emails and 5% of genuine emails. An email is flagged. What is the probability it is spam? Give 3 decimal places. → 0.873Prompt: Find the total probability of being flagged (two branches).

Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-11.