Handling Data · Chance or Probability · Lesson 2
Classifying the Likelihood of Outcomes with Dice
A die, a coin and a spinner are the tools of probability experiments. In this lesson you will list every possible outcome of an experiment (the sample space) and use it to decide whether a given outcome is impossible, possible or certain, first with one die and then with two dice.
By the end you will be able to
- ✔List the sample space of a single experiment such as rolling a die or tossing a coin.
- ✔Classify an outcome of throwing one die as impossible, possible or certain.
- ✔Build the table of totals for two dice and use it to classify totals.
- ✔Explain the reason for each classification.
Key words
- Experiment
- An activity with results we cannot be sure of beforehand, such as throwing a die.
- Outcome
- One possible result of an experiment, for example rolling a 4.
- Sample space
- The list of ALL possible outcomes of an experiment.
- Fair die
- A die whose six faces, numbered 1 to 6, are equally likely to land on top.
Curriculum reference
Content standard: B7.4.2.1 Identify the sample space for a probability experiment involving single events and express the probabilities of given events as fractions, decimals, percentages and/or ratios to solve problems.
Indicator: Classify the likelihood of a single outcome occurring in a probability experiment as impossible, possible, or certain.
Starter
The ludo and ayo players' question
In a game of ludo at the Cape Coast beach, Kwame needs a 6 to leave his base. Ama says, 'You can never get a 7 with one die, but with two dice you can get 7 many ways!' Who is right? To settle the argument, we must first write down everything that can happen. That list is the secret to all of probability.
1. Experiments, outcomes and the sample space
An experiment is any action whose result we cannot be sure of beforehand: tossing a coin, throwing a die or spinning a spinner. Each possible result is an outcome. The list of all possible outcomes is the sample space.
| Experiment | Sample space | Number of outcomes |
|---|---|---|
| Toss a coin | {Head, Tail} | 2 |
| Throw one die | {1, 2, 3, 4, 5, 6} | 6 |
| Spin a spinner with 8 equal sectors numbered 1 to 8 | {1, 2, 3, 4, 5, 6, 7, 8} | 8 |
| Pick one card from cards A, B, C, D, E | {A, B, C, D, E} | 5 |
The sample space is your checklist. To classify an outcome, look at the sample space and ask: 'Is my outcome in the list?' If it is in the list, it is possible. If it is not in the list, it is impossible. If it is every outcome (or the event covers the whole list), it is certain.
2. Classifying outcomes of throwing one die
A fair die has six faces numbered 1, 2, 3, 4, 5, 6, so the sample space is {1, 2, 3, 4, 5, 6}. Look at the curriculum events:
| Outcome | In the sample space? | Classification | Reason |
|---|---|---|---|
| Obtaining the number 1 | Yes | Possible | 1 is one of the six faces, but five other numbers could come up |
| Obtaining the number 7 | No | Impossible | The faces only go up to 6 |
| Obtaining the number 4 | Yes | Possible | 4 is a face, but it will not come up every time |
| Obtaining a number less than 7 | All six outcomes | Certain | Every face (1 to 6) is less than 7 |
| Obtaining a number greater than 6 | None of them | Impossible | No face is greater than 6 |
▶ A spinner with six equal sectors behaves like a fair die. Only one sector is favourable for 4, so it is possible but not certain.
3. Classifying totals with two dice
When two dice are thrown together and the two numbers are added, there are pairs of results. The table shows the total for every pair. This table is the sample space for the total.
▶ Look for the total you are asked about. If it is in the table it is possible; if the table has no such total, it is impossible.
| Total | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Ways to get it | 1 | 2 | 3 | 4 | 5 | 6 | 5 | 4 | 3 | 2 | 1 |
From the table: a total of 12 is possible (only one way: 6 + 6). A total of 2 is possible (only one way: 1 + 1). A total of 13 is impossible since the biggest total is 6 + 6 = 12. A total of 1 is also impossible since the smallest total is 1 + 1 = 2. A total that is at least 2 and at most 12 is certain.
4. A method for any experiment
Let us practise the thinking with a Ghanaian example. Naa has a bag with ten tiles numbered 1 to 10 and picks one without looking. The sample space is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Picking a 12 is impossible because 12 is not a tile. Picking a number from 1 to 10 is certain because every tile fits. Picking a 5 is possible because there is a tile numbered 5, but nine other tiles could also be picked. Notice that we never need to know how many times a tile will be chosen. We only need to know what is on the tiles.
It is also helpful to see that 'impossible' and 'certain' are two ends of the same idea. If the event 'picking a number greater than 10' is impossible, then its opposite, 'picking a number that is 10 or less', is certain. Whenever you find one, you have also found the other.
- 1Write down the experiment and list the sample space.
- 2Write down the outcome (or event) in the question.
- 3Check: is the outcome in the sample space? Is it the only outcome? Does it cover all outcomes?
- 4Classify as impossible, possible or certain, and write the reason.
Always give a reason. A teacher marking your work wants to see 'impossible, because a die has no face with 7', not just the word 'impossible'.
Worked examples
Try each one yourself first, then open the solution and compare every step.
Example 1
One die
▶ Only one of the six sectors is favourable for 1, so it is possible but not certain.
Show the solution
- Step 1Sample space = {1, 2, 3, 4, 5, 6}.
- Step 2(a) 1 is in the sample space, along with others: possible.
- Step 3(b) 7 is not in the sample space: impossible.
- Step 4(c) 4 is in the sample space, along with others: possible.
Answer: (a) possible, (b) impossible, (c) possible
Example 2
A certain outcome
Show the solution
- Step 1Check each outcome: 1, 2, 3, 4, 5 and 6 are all less than 7.
- Step 2Every outcome gives the result, so the outcome is certain.
Answer: Certain
Example 3
Two dice
Show the solution
- Step 1Smallest total = 1 + 1 = 2; largest total = 6 + 6 = 12.
- Step 2(a) 12 = 6 + 6 can happen but is not sure: possible.
- Step 3(b) 2 = 1 + 1: possible.
- Step 4(c) 13 is bigger than the largest total 12: impossible.
Answer: (a) possible, (b) possible, (c) impossible
Example 4
How many ways?
Show the solution
- Step 1List the pairs: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2).
- Step 2Count them: 5 ways.
Answer: 5 ways
Example 5
Cards
Show the solution
- Step 1Sample space = {A, B, C, D, E}.
- Step 2(a) F is not in the sample space: impossible.
- Step 3(b) C is one of five: possible.
- Step 4(c) Every outcome is a letter from A to E: certain.
Answer: (a) impossible, (b) possible, (c) certain
Summary
- The sample space is the list of all possible outcomes of an experiment.
- An outcome not in the sample space is impossible; one in the list is possible; one that covers the whole list is certain.
- A die has the sample space {1, 2, 3, 4, 5, 6}, so 7 is impossible and 'less than 7' is certain.
- Two dice have 36 pairs and totals from 2 to 12, so 13 is impossible and 12 and 2 are possible.
- Always give a reason for your classification.
Practice: 15 questions
5 easy, 5 medium, 5 hard. Each one is marked straight away, with a full worked solution.
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