At a glance. One lesson of about 60 minutes. It assumes fractions and simple probability from Year 8 and the notation n(A) from Lessons 1 and 2. Students build sample spaces with tables, use P(A) = n(A) ÷ n(S) and P(A′) = 1 − P(A).
Calculator: no calculator. None is needed. Counting cells and simplifying small fractions is the reasoning, and the probabilities are exact fractions or terminating decimals.
What students will learn
Know
- P(A) = n(A) ÷ n(S) when the outcomes are equally likely.
- 0 ≤ P(A) ≤ 1, P(S) = 1 and P(A′) = 1 − P(A).
- A table of two stages lists every outcome once.
Understand
- The sample space must contain outcomes that are equally likely.
- Order matters when the stages are different, so (1, 4) and (4, 1) are different outcomes.
- Counting the complement is often quicker.
Do
- Draw a table for two stages and read an event from it.
- Write a probability as a fraction, a decimal and a percentage.
- Solve a simple equation from P(A) + P(A′) = 1.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: sample spaces and the probability of an event and its complement (questions 1 to 8).
- Criterion B. Investigating patterns: the first figure. Ask students why the total 7 is the most likely and which totals have the same probability.
- Criterion C. Communicating: writing out a sample space so that another student can check it (the faded example and question 4).
- Criterion D. Applying mathematics in real-life contexts: the stall owner's claim and what a fair payout would be.
The one thing to do with the figure
Make students commit before they slide. In the first figure, ask “How many of the 36 cells have a total of 7?” and take written answers before the slider is moved to that event. Many students say 1 or 2.
In the second figure, set 24 tickets to 0 and then to 24 and ask what the probabilities are called: impossible and certain.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: the stall owner's claim. Collect predictions and reasons; do not settle them yet. |
| 10 min | First figure and key idea. Settle the hook by counting the cells. |
| 10 min | Second figure, the warning card and the worked example; pairs finish the faded example. |
| 5 min | The check. If many choose A, return to the coins and list all eight outcomes. |
| 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 | 1/4; 35%; 12. |
|---|---|
| Hook | No. The totals are not equally likely: 7 can be made 6 ways of the 36, so P(7) = 1/6; 2 and 12 can each be made one way. |
| Worked example | P(prime) = 2/5 and P(not prime) = 3/5. |
| Faded example | P(sum is 4) = 3/9 = 1/3. |
| 1. Equally likely outcomes | B. A treats the four totals as equally likely; C treats happens or not as 50:50; D answers a different question. |
| 2. Counting outcomes in a table | 4. (1, 4), (2, 3), (3, 2), (4, 1). |
| 3. A complement as a decimal | 0.55. 11/20. |
| 4. Counting in a two-dice table | 6. Differences of 4: 4 outcomes; of 5: 2 outcomes. |
| 5. Probability as a percentage | 80%. 20/25. |
| 6. Using P(A) + P(A′) = 1 | 0.2. 4x + 0.2 = 1. |
| 7. Complement in a table | 0.75. 1 − 9/36. |
| 8. Counting a maximum | 5. The pairs from {1, 2, 3} with at least one 3. |
| 9. Size of a sample space | C. A leaves out the coin; B adds instead of multiplying; D treats the coin as six-sided. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Counts totals instead of outcomes, treats a two-way split as even, or answers a different question. | List the 8 outcomes HHH to TTT on the board and ask which are three heads. |
| 2 or 5 in question 2 | Leaves out the reversed pair, or gives the total. | Use a coloured pen for the first die and another for the second, so order is visible. |
| 0.45, 0.3 or 0.25 in question 3 | Gave P(red) or a single colour. | Ask: which counters are not red? Count all of them. |
| 3, 4 or 12 in question 4 | Forgot the mirror pairs, took exactly 4 only, or used at least 3. | Shade the cells on the table and count them. |
| 20, 36 or 44 in question 5 | Gave P(cycle) or a single group. | Write the groups as 9 + 11 + 5 = 25 and shade the ones that are not cycling. |
| 0.25, 0.4 or 0.6 in question 6 | Ignored the 0.2, or gave P(A) or P(A′) instead of x. | Substitute x = 0.2 back to check: 0.4 + 0.6 = 1. |
| 0.25 or 0.5 in question 7 | Gave the odd probability, or the chance of one die. | Make a table with E for even and O for odd; count the cells with no E. |
| 7, 6 or 3 in question 8 | Included a 4, double counted (3, 3), or missed the second die. | Write the pairs in a grid and tick the ones where the larger is 3. |
| A, B or D in question 9 | Left out the coin, added, or treated the coin as six-sided. | Draw a tree with the two coin branches, then six die branches from each. |
Questions worth asking
- Why are (1, 4) and (4, 1) different outcomes with two dice? Listen for “one die is first and the other is second, so each is a separate result”.
- Why is a sum of 7 the most likely total? More pairs add to 7 than to any other total: 1 and 6, 2 and 5, and so on, in both orders.
- Why is it sometimes easier to find the complement? Fewer cases to count: the product of two numbers is even in almost every case, but odd only when both dice are odd, 9 of 36.
- Can a probability be more than 1? Less than 0? No: n(A) is between 0 and n(S).
Differentiation
A way in for anyone stuck
- Give a blank 6 by 6 grid, and have students colour the cells of each event.
- Write “equally likely?” at the top of every sample space and ask the question each time.
- Use two colours of dice, or ask students to say “red die first, blue die second”.
Stretch
- A game: Mia wins if the sum is 7 and Omar wins if it is 6 or 8. Who is more likely to win, and by how much? Design a fair payout.
- How many outcomes are there for four coins? For n coins? What is P(all heads)?
Students learning in English as an additional language
“Outcome” is one result, “event” is a group of outcomes you care about, and “sample space” is the list of everything that can happen. “Fair” means every outcome is equally likely. “Complement” is the “not” event.
| Word | Plain meaning | Example |
|---|---|---|
| outcome | one possible result of an experiment | the pair (2, 5) on two dice |
| sample space | the list of all possible outcomes, written S | the six faces 1 to 6 of one die |
| event | a set of outcomes you are interested in | the dice show a double |
| equally likely | each outcome has the same chance of happening | each side of a fair die |
| probability | a number from 0 to 1 that says how likely an event is | 1/6 for rolling a six |
| complement | the event that A does not happen, written A′ | not rolling a six |
Sentence starters for explaining aloud or in writing:
- The sample space has … equally likely outcomes because ….
- The event A contains … of them, so P(A) = ….
- I used the complement because ….
Two things not to say
“It is either yes or no, so it is 50:50.” Two outcomes need not be equally likely.
“Probability is just a fraction.” It is a fraction of equally likely outcomes; the list must be complete first.
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.
- TableTwo fair coins are tossed. List the sample space and find P(exactly one head).
{HH, HT, TH, TT}; 2/4 = 1/2. - ComplementA card is chosen at random from 20 numbered 1 to 20. P(not a multiple of 5)?
16/20 = 4/5.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA spinner in a Mumbai arcade has 12 equal sectors and 5 of them win a prize. What is P(no prize)?
7/12. - Earlier work (Lesson 2)n(A) = 11, n(B) = 8 and n(A ∩ B) = 3. Find n(A ∪ B).
16.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningExplain why the sum of two dice is more likely to be 6 than 3.
Six has 5 pairs and 3 has only 2: (1, 2) and (2, 1). - Working backwardsP(A) = 0.35 in a sample space of 40 equally likely outcomes. How many outcomes are in A?
14.
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? Give pairs two dice and a blank table and have them tally 36 rolls, then compare with the 36 cells.