At a glance. One lesson of about 60 minutes. It follows independence. Students compute P(B | A) from counts and from probabilities, read the second stage of a tree, use the multiplication rule, and explain why P(A | B) and P(B | A) differ.
Calculator: no calculator. No calculator is needed. The counts are chosen so that each division is a short decimal, and the lesson is about choosing the right group to divide by.
What students will learn
Know
- P(B | A) = P(A ∩ B) ÷ P(A) = n(A ∩ B) ÷ n(A).
- P(A ∩ B) = P(A) × P(B | A).
- A and B are independent exactly when P(B | A) = P(B).
Understand
- Knowing an event has happened changes the sample space, and so the probability.
- Trees, tables and Venn diagrams are three views of the same shrinking of the sample space.
- The order of the condition matters: the same intersection is divided by a different group.
Do
- Calculate a conditional probability from a table, a Venn diagram or a tree.
- Use the multiplication rule and the total over the routes of a tree.
- Say in words what P(A | B) and P(B | A) each mean.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: calculating conditional probabilities and using the multiplication rule (questions 1 to 7).
- Criterion B. Investigating patterns: the figure and the what-if. Students compare P(B | A) with P(B) and link equality to independence.
- Criterion C. Communicating: reading P(B | A) aloud as a sentence and writing the group that is given at the bottom of the fraction.
- Criterion D. Applying mathematics in real-life contexts: the homework and sport hook, where the chance for a sub-group differs from the whole.
The one thing to do with the figure
Ask for a prediction before each slide. Before anyone reads the readout, ask: “Will this be more or less than P(B)?” Then slide to P(A | B) and ask why the same 48 gives a different answer.
Ask students to name the group that is shaded each time, and to say it as the denominator before reading the readout.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: team sport and homework. Collect predictions. |
| 12 min | The figure with predictions; the key idea; settle the hook. |
| 10 min | Warning card and worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A or C, return to the shaded group in the figure. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 0.1; 4; 0.45. |
|---|---|
| Hook | Different. More than an hour is more likely for the team sport players (0.48 against 0.4 for everyone); turned round, team sport is more likely among the students with more than an hour (0.6 against 0.5). |
| Worked examples | 0.45; 2/15; 3/10 and 3/8. |
| Faded example | 0.22. |
| 1. Conditional probability from the definition | B. A is the intersection itself; C multiplies; D divides the wrong way round. |
| 2. Conditional probability from a count | 0.35. 21 ÷ 60. |
| 3. Second stage of a tree without replacement | 0.3. 3 ÷ 10. |
| 4. The multiplication rule | 0.36. 0.8 × 0.45. |
| 5. Reversing the condition | 0.25. 0.15 ÷ 0.6. |
| 6. Conditional probability on a complement | 0.55. 88 ÷ 160. |
| 7. Solving for the size of a bag | 5. n² + 7n − 60 = 0. |
| 8. P(A | B) against P(B | A) | A. B swaps the two; C gives the intersection; D gives the unconditional values. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Gave the intersection, multiplied, or divided the wrong way round. | Say: the bottom is the event we are given. |
| 0.14 or 0.4 in question 2 | Divided by everyone, or gave the share with a camera. | Count only the group that is given. |
| 0.27 or 0.33 in question 3 | Left the taken counter in the total, or left out the green counters. | Cross the counter out and recount. |
| 0.45, 0.56 or 1.25 in question 4 | Gave the conditional probability, divided, or added. | Write P(A ∩ B) = P(A) × P(B | A) and compare. |
| 0.45 or 4 in question 5 | Subtracted, or divided the wrong way round. | Circle the given event; it is the bottom. |
| 0.44, 0.88 or 0.3 in question 6 | Divided by everyone, by n(A), or used B as the condition. | Draw the table and shade the group B′. |
| 9 or −12 in question 7 | Gave the total, or kept the negative root. | Reject a negative count of counters. |
| B, C or D in question 8 | Swapped the conditionals, gave the intersection, or gave the unconditional values. | Write the denominator under each one first. |
Questions worth asking
- Why does the sample space shrink? We know the event has happened, so only outcomes inside it remain possible.
- On a tree, what do the second-stage branches show? Conditional probabilities: the chance of the second event given the first-stage outcome.
- Why can P(A | B) and P(B | A) be very different? They divide the same intersection by two different groups, which can have very different sizes.
- When are P(B | A) and P(B) equal? Exactly when A and B are independent.
Differentiation
A way in for anyone stuck
- Give a two-way table frame and ask students to shade the group that is given before writing any fraction.
- Use the sentence frame: “Out of the … who are …, … are ….”
- Use counts out of 100 before moving to decimals.
Stretch
- Show that for any events, P(A | B) × P(B) = P(B | A) × P(A), and say what that means for a tree drawn in two orders.
- A family has two children, and at least one is a girl. What is the probability that both are girls? Compare with “the elder is a girl”.
Students learning in English as an additional language
“Given” means “we know that”. P(B | A) is read “the probability of B given A”. “Conditional” means that the probability depends on a condition. “Without replacement” means that the item taken is not put back.
| Word | Plain meaning | Example |
|---|---|---|
| conditional probability | the probability of an event when another has happened | the chance a bus is late given that it is raining |
| given | the event we know has happened, written after the bar | | in P(B | A) the given event is A |
| reduced sample space | the outcomes that remain once the given event is known | only the walkers, when we know the student walks |
| multiplication rule | P(A ∩ B) = P(A) × P(B | A) | multiplying along a route of a tree |
| without replacement | the item taken is not put back, so the next probabilities change | taking two cards from a pack |
| two-way table | a table of counts for two categories | walk or not against glasses or not |
Sentence starters for explaining aloud or in writing:
- Given that …, the sample space is …
- P(B | A) = … ÷ …, because the group is …
- P(A | B) is different from P(B | A) because …
Two things not to say
“P(A | B) and P(B | A) are the same.” They share the intersection but divide by different groups.
“Divide by the total.” Divide by the group that is given.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- CountOf 80 people, 30 wear a hat, and 6 of the hat wearers also wear sunglasses. Find P(sunglasses | hat).
6 ÷ 30 = 0.2. - Earlier work (lesson 4)A and B are independent, P(A) = 0.4 and P(B) = 0.3. Find P(A ∩ B).
0.12.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextIn a class, P(sleeps 8 hours) = 0.6, and 0.45 of the class sleeps 8 hours and eats breakfast. Find P(breakfast | sleeps 8 hours).
0.45 ÷ 0.6 = 0.75. - TreeA box has 5 white and 5 black counters. Two are taken without replacement. Find P(both black).
5/10 × 4/9 = 20/90 = 2/9.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningP(B | A) = 0.5, P(B | A′) = 0.2 and P(A) = 0.4. Find P(B).
0.4 × 0.5 + 0.6 × 0.2 = 0.2 + 0.12 = 0.32. - ProofProve that if P(B | A) = P(B), then P(A ∩ B) = P(A) × P(B).
P(A ∩ B) = P(A) × P(B | A) by the multiplication rule, and P(B | A) = P(B).
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to the slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No calculator is needed. If class data is used, gather it anonymously.