At a glance. One lesson of about 60 minutes. Theoretical probability as favourable over total for equally likely outcomes, and P(not A) = 1 − P(A).
Calculator: no calculator. The denominators are 6, 8, 10, 16, 20, 36 and 40, so every probability can be written as a fraction without a calculator.
What students will learn
Know
- P(A) is favourable outcomes over all outcomes, for equally likely outcomes.
- P(not A) = 1 − P(A).
- The probabilities of all the outcomes add to 1.
Understand
- The formula needs outcomes that are equally likely.
- Counting the complement can be quicker than counting the event.
Do
- Find a probability from a list or a table of equally likely outcomes.
- Use the complement.
- Work back from a probability to a count.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: using the formula and the complement (questions 2 to 5).
- Criterion B. Investigating patterns: with the figure, describe how the number of cells changes with the total and explain the symmetry about 7.
- Criterion C. Communicating: writing probabilities as simplified fractions and justifying that outcomes are equally likely.
- Criterion D. Applying mathematics in real-life contexts: games, raffles and bags, and judging when the formula does not apply.
The one thing to do with the figure
Ask for the most and least likely totals before the first move. Many students predict a flat distribution. Walk the slider from 2 to 12 and ask them to describe the shape of the counts in words (1, 2, 3, 4, 5, 6, 5, 4, …).
Then ask why 6/36 is not 6 ÷ 11. The grid is the answer: each cell is equally likely, the total is not.
A possible order
| 5 min | Warm-up, then answers. |
|---|---|
| 10 min | Hook: which total? Record predictions on the board. |
| 15 min | The figure; the formula and the complement. |
| 10 min | Worked examples; the cards in pairs. |
| 5 min | Question 1, the check. |
| 10 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 8; 1/6; 20% and 20. |
|---|---|
| Worked examples | Counters: 3/8 and 5/8. Die: 1/2. Coins: 7/8. |
| Faded example | 1/5 (0.2). |
| 1. Favourable over total | B, 1/3. A compares blue with the others, not with everything. |
| 2. Greater than | 0.3 = 3/10. |
| 3. Multiples | 0.125 = 5/40. |
| 4. Two dice, a total at least 6 | 0.375 = 6/16. |
| 5. Three colours | 0.45. |
| 6. Equally likely? | A. B, C and D count results that are not equally likely. |
| 7. Missing counters | 4 green counters. |
| 8. Working back to the total | 15 counters. |
| 9. Greatest probability | C. Counting favourable cards for each event (4, 5, 7, 3) settles it. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| 1/2 or 1/4 in question 1 | Used non-favourable outcomes or the count as the denominator. | Ask “out of how many in total?” and draw the bag. |
| 0.7 or 0.4 in question 2 | Counted the complement, or included the boundary value. | List the favourable faces on the board. |
| 5, 0.2 or 0.875 in question 3 | Gave the count, used the divisor 8 as the count, or counted the tickets that are not multiples. | List 8, 16, 24, 32, 40 and count them out of 40. |
| 0.25 in question 4 | Treated (2, 4) and (4, 2), and (3, 4) and (4, 3), as single outcomes. | Use two colours of dice, then return to the grid. |
| 0.55 in question 5 | Gave P(red or blue) instead of P(green). | Draw all three on a 0 to 1 line. |
| 10 or 2.4 in question 8 | Used P(not red), or multiplied instead of dividing. | Check: 0.4 of 10 is 4, not 6. |
Questions worth asking
- Why isn't the probability of a total of 7 equal to 1/11? Listen for “the totals are not equally likely”.
- When is it quicker to use 1 − P(A)? When the event has many outcomes and the complement has few: at least one head.
- Can P(A) and P(not A) both be more than 1/2? No: they add to 1. Ask for a reason.
Differentiation
A way in for anyone stuck
- Give the 6 by 6 grid on paper, already filled with the totals.
- Let students colour all cells of one total before using the figure.
- Work with small bags of counters before the dice.
Stretch
- Two fair dice are rolled. Find P(the total is a prime number). (15/36 = 5/12.)
- A bag has red and blue counters and P(red) = 3/7. What is the smallest number of counters it could hold? (7.)
Students learning in English as an additional language
Pair each word with an action: pointing to the shaded cells for “favourable”, sweeping across the whole grid for “all outcomes”.
| Word | Plain meaning | Example |
|---|---|---|
| theoretical probability | the probability worked out from equally likely outcomes | a die shows 5 with probability 1/6 |
| favourable outcome | an outcome that makes the event happen | 8, 9 and 10 for “greater than 7” |
| equally likely | each outcome has the same chance as every other | the faces of a fair die |
| complement | the event not happening | not red |
| fair | with no outcome favoured over any other | a fair coin |
Sentence starters for explaining aloud or in writing:
- The probability is … out of …, which simplifies to …
- I can use the formula because the outcomes are / are not equally likely.
- P(not …) is 1 minus …, which is …
Two things not to say
“Just put the number you want over the total.” Without the equally-likely check students put 1 over 11 for dice totals.
“Probability is a fraction.” It is a number from 0 to 1, and it can be written as a fraction, a decimal or a percentage.
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.
- CountersA bag has 2 red, 3 blue and 5 green counters. What is P(green)?
5/10 = 1/2. - DieWhat is the probability of an even number on a fair die?
3/6 = 1/2.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ComplementP(win) = 0.35. What is P(not win)?
0.65. - Earlier work (trees)A coin is tossed 3 times. What is the probability of exactly 3 heads?
1/8.
3Challenge
Reasoning, generalizing and working backwards.
- CardsA card is picked from 1 to 30. What is P(multiple of 6)?
5/30 = 1/6. - MissingP(blue) = 0.3 and there are 9 blue counters. How many counters in all?
30.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The slider works with the keyboard and by touch; “Show me the steps” gives four cases in words.
On paper: a 6 by 6 grid and a highlighter do the same job as the figure.