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Math lesson plan · 11th grade (Year 12 in England) · 45 minutes

Use P(A | B) = P(A and B) ÷ P(B)

Statistics and probability

CCSS HSS.CP.A.3CCSS HSS.CP.B.6England: A-level Statistics

Objective and success criteria

Students will: Use P(A | B) = P(A and B) ÷ P(B).

  • 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: Conditional probability and independence

Leads to: Reverse conditional probability with a tree (tests and filters)

Materials: Squared paper for charts, Class survey data, Sticky notes for a live chart, Mini whiteboards and pens, Printed student sheet (below).

Warm-up · 5 min

  • Quick retrieval from earlier learning (Conditional probability and independence). Students answer on mini whiteboards.
  • Show answers together and note who needs support in the main task.
  1. Survey: Likes science & is in Club — 25; Likes science & not in Club — 11; Doesn't like science & in Club — 8; neither — 10. Given a student is in the Club, what is the probability they like science? Simplest form. → 25/33Prompt: 'Given' means only look at the Club members.
  2. P(A) = 0.3, P(B) = 0.5 and P(A and B) = 0.15. Are A and B independent? (Yes / No) → YesPrompt: A and B are independent if P(A and B) = P(A) × P(B).
  3. Survey: Likes science & is in Club — 10; Likes science & not in Club — 5; Doesn't like science & in Club — 11; neither — 5. Given a student is in the Club, what is the probability they like science? Simplest form. → 10/21Prompt: 'Given' means only look at the Club members.

Hook · 5 min

  • Take a quick class survey (favorite fruit, travel to school) and ask what the results tell us.
  • Share the objective: Use P(A | B) = P(A and B) ÷ P(B). Read the success criteria together.

I do: model it · 10 min

  • Work through the example out loud, saying why you do each step, not just what you do.
  • Example: P(B) = 0.5 and P(A and B) = 0.1. Find P(A | B).
  • Step 1: 0.1 ÷ 0.5 = 0.2
  • Answer: 0.2
  • Pause after each step and ask a student to say what comes next.

We do: guided practice · 10 min

  • Do these together. Students try each step on whiteboards before you reveal it.
  • Use the hints as prompts rather than giving the answer.
  1. P(B) = 0.5 and P(A and B) = 0.2. Find P(A | B). → 0.4Prompt: P(A | B) = P(A and B) ÷ P(B).
  2. The probability it rains tomorrow is 0.7. If it rains, the probability the bus is late is 0.75. What is the probability that it rains and the bus is late? → 0.525Prompt: P(A and B) = P(B) × P(A | B).

You do: independent practice · 10 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 · 5 min

  • Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
  1. Phone survey: Android: under 18 22, 18+ 17 iPhone: under 18 26, 18+ 25 A person aged 18+ is chosen at random. Find P(iPhone | 18+) as a fraction in simplest form. → 25/42Prompt: Only look at the 18+ column.
  2. 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).

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: Conditional probability and independence.
  • 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

  • Move on to: Reverse conditional probability with a tree (tests and filters).
  • 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

eurekami

Use P(A | B) = P(A and B) ÷ P(B)

Name

Date

Score/ 8

  1. Question 1

    P(B) = 0.8 and P(A and B) = 0.2. Find P(A | B).

    Answer

  2. Question 2

    The probability it rains tomorrow is 0.4. If it rains, the probability the bus is late is 0.8. What is the probability that it rains and the bus is late?

    Answer

  3. Question 3

    Phone survey: Android: under 18 21, 18+ 10 iPhone: under 18 16, 18+ 6 A person aged 18+ is chosen at random. Find P(iPhone | 18+) as a fraction in simplest form.

    Answer(as a fraction)

  4. Question 4

    P(B) = 0.4 and P(A and B) = 0.08. Find P(A | B).

    Answer

  5. Question 5

    The probability it rains tomorrow is 0.6. If it rains, the probability the bus is late is 0.8. What is the probability that it rains and the bus is late?

    Answer

  6. Question 6

    Phone survey: Android: under 18 14, 18+ 17 iPhone: under 18 22, 18+ 5 A person aged 18+ is chosen at random. Find P(iPhone | 18+) as a fraction in simplest form.

    Answer(as a fraction)

  7. Question 7

    P(B) = 0.8 and P(A and B) = 0.48. Find P(A | B).

    Answer

  8. Question 8

    The probability it rains tomorrow is 0.3. If it rains, the probability the bus is late is 0.5. What is the probability that it rains and the bus is late?

    Answer

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Answer key: Use P(A | B) = P(A and B) ÷ P(B)

  1. 1. 0.25

  2. 2. 0.32

  3. 3. 3/8

  4. 4. 0.2

  5. 5. 0.48

  6. 6. 5/22

  7. 7. 0.6

  8. 8. 0.15