At a glance. One lesson of about 60 minutes. It assumes P(A) = n(A) ÷ n(S) and the union formula for counts. Students read probabilities from two-way tables, find a probability within a group, use P(A ∪ B) = P(A) + P(B) − P(A ∩ B), and decide when events are mutually exclusive.
Calculator: no calculator. None is needed. Every table is small and every fraction can be simplified by hand; the reasoning is in choosing the bottom number.
What students will learn
Know
- P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
- Mutually exclusive events have P(A ∩ B) = 0.
- Within a group, the group's total is the bottom number.
Understand
- The same top number can sit over different bottom numbers depending on the group.
- A table, a Venn diagram and a formula carry the same information.
- Mutually exclusive is a special case of the addition rule.
Do
- Read single, joint, either-or and within-group probabilities from a table.
- Use the addition rule with probabilities and with counts.
- Decide whether two events are mutually exclusive.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: tables, Venn diagrams and the addition rule (questions 1 to 8).
- Criterion B. Investigating patterns: the second figure. Ask students to predict the union before moving the overlap, and what happens to the formula at 0.
- Criterion C. Communicating: stating the group before writing a fraction (question 3 and question 5, and the hook's two sentences).
- Criterion D. Applying mathematics in real-life contexts: how claims about a group can be true or false depending on which group is meant.
The one thing to do with the figure
Make students commit before they slide. In the first figure, ask “Will P(chess, among females) be larger or smaller than P(female and chess)?” before moving. The numerator is the same; many say it must be the same.
In the second figure, slide the overlap to 0 and ask what the formula becomes.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: the two coaches. Collect views on whether both can be right. |
| 10 min | First figure and key idea: six probabilities from one table. Settle the hook. |
| 10 min | Second figure, the warning card and the worked example; pairs finish the faded example. |
| 5 min | The check. If many choose A or B, return to the pair with a shared card. |
| 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 9. |
The answers
| Warm-up | 2/5; 19; 75%. |
|---|---|
| Hook | Both can be right. In the figure, 20 of the 29 chess players are female, but only 20 of the 45 females play chess. |
| Worked example | P(bus) = 13/24; P(Year 11 and bus) = 1/6; among Year 11, 2/5. |
| Faded example | P(Year 10 or bus) = 90/120 = 3/4. |
| 1. Mutually exclusive events | C. A, B and D each have a card that is in both events (10, 2 and 9). |
| 2. A percentage from a table | 52%. 19 + 33 out of 100. |
| 3. Probability within a group | 62%. 31 out of 50. |
| 4. Neither, from P(A), P(B) and P(A and B) | 0.25. 1 − 0.75. |
| 5. Within a group, from totals | 60%. 36 out of 60. |
| 6. Deciding mutually exclusive from probabilities | A. B confuses the union with the whole space; C uses size; D uses a test that overlapping events can also pass. |
| 7. A union with a complement | 0.8. 1 − (0.3 − 0.1). |
| 8. Mutually exclusive outcomes covering everything | 0.25. 2x + 0.5 = 1. |
| 9. The addition rule | B. A forgets the overlap; C adds it; D multiplies. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Does not search for a shared outcome, or checks only one card. | Ask for a card that is in both events. If one exists, they are not mutually exclusive. |
| 19, 33 or 48 in question 2 | Used only one group, or answered the opposite question. | Underline “did not ride” and find both numbers in the survey. |
| 31 or 48 in question 3 | Used the whole survey as the bottom number. | Cross out everyone who is not a member before choosing the bottom number. |
| 0.05, 0.45 or 0.75 in question 4 | Double counted the overlap, complemented one set, or gave the union. | Draw a two-circle diagram and fill the overlap first. |
| 40 or 24 in question 5 | Counted out of the wrong group. | Ask: you are told the person is in the chess club, so how many people could they be? |
| B, C or D in question 6 | Confuses mutually exclusive with the union being certain, with size, or with a sum below 1. | Ask for a card or outcome that is in both. If none, then exclusive. |
| 1.2, 0.2 or 0.9 in question 7 | Added without the overlap, gave the complement part, or complemented the overlap. | Shade A ∪ B′ on a diagram: which region is not shaded? |
| 0.5, 0.15 or 0.35 in question 8 | Stopped after finding a probability and did not return to x. | Re-read the question: it asks for x, not for P(red). |
| A, C or D in question 9 | Forgot the overlap, added it, or multiplied. | Write P(piano) and P(violin) on a diagram, with the overlap in the middle. |
Questions worth asking
- Why is P(A ∩ B) = 0 for mutually exclusive events? Listen for “if they cannot happen together, there is no outcome in both”.
- Can two events with probability 0.6 each be mutually exclusive? No: their probabilities would add to 1.2, which is more than 1.
- Does “among females” change the numerator or the denominator? The denominator: the group is the new total; the numerator stays as the people in both.
- Are mutually exclusive events independent? No, unless one of them has probability 0: if one happens the other cannot, so one changes the chance of the other.
Differentiation
A way in for anyone stuck
- Give the table with row and column totals already filled in, and have students colour the cells they use.
- Write the group in a box above every fraction: “out of ___ people”.
- Use a printed two-circle diagram for the addition rule, and have students write the counts in each region.
Stretch
- Show that if A and B are mutually exclusive and both have probability greater than 0, they cannot be independent.
- Three events A, B and C are mutually exclusive with P(A) = 0.3 and P(B) = 0.45. What values can P(C) take?
Students learning in English as an additional language
“Among” and “given that” mean “only count this group”. “Mutually exclusive” means that if one happens, the other cannot. “Either … or” includes both unless the question says “exactly one”. Underline the group in every sentence before writing a fraction.
| Word | Plain meaning | Example |
|---|---|---|
| two-way table | a table that classifies the same group by two features at once | pupils by year group and how they travel |
| within a group | counting only the people or outcomes in one named group | chess players among the females |
| mutually exclusive | two events that cannot both happen at the same time | a coin landing heads and landing tails on one toss |
| union | outcomes in A or in B or in both, written A ∪ B | a card that is odd or greater than 15 |
| intersection | outcomes in both A and B, written A ∩ B | a card that is a square and also odd |
| total | the number in a whole row, column or group | all 120 pupils |
Sentence starters for explaining aloud or in writing:
- I am counting out of … because the question says ….
- The events are / are not mutually exclusive because ….
- I subtract P(A ∩ B) because ….
Two things not to say
“Just use the total.” The bottom number changes when the question names a group.
“Mutually exclusive means independent.” If A happens, B cannot, so one event changes the chance of the other.
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.
- Table30 pupils study French, 20 study German and 5 study both, out of 60. How many study neither?
60 − 45 = 15. - ExclusiveA die is rolled. Are “a number greater than 4” and “an even number” mutually exclusive?
No: a 6 is both greater than 4 and even.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextIn a city tour of 40 people in Marrakesh, 24 visited the souk, 18 visited the gardens and 6 visited both. What is P(neither)?
40 − 36 = 4, so 4/40 = 1/10. - Earlier work (Lesson 4)A bag holds 5 red and 3 blue counters. Two are taken without replacement. P(both red)?
5/8 × 4/7 = 20/56 = 5/14.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningP(A) = 0.6 and P(B) = 0.7. Explain why A and B cannot be mutually exclusive.
Mutually exclusive would need P(A ∪ B) = 1.3, which is more than 1. - Working backwardsP(A) = 0.5, P(A ∪ B) = 0.8 and P(A ∩ B) = 0.15. Find P(B).
0.8 = 0.5 + P(B) − 0.15, so P(B) = 0.45.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to the slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Draw the table on the board and have students come up to circle the cells for each probability.