Systematic possibilities and equally likely outcomes · Foundation
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| sample space/ˈsæmpl speɪs/ | 样本空间 | yàng běn kōng jiān |
Two fair dice can total seven in six different ways, but total two in only one way. The possible totals are not equally likely.
- Two fair dice can total seven in six different ways, but total two in only one way. The possible totals are not equally likely.
- This lesson studies sample space 样本空间: The complete set of outcomes in an experiment.
Choose the mathematical structure
- List outcomes systematically using a grid, table or tree. For equally likely outcomes, probability is favourable outcomes/total outcomes. Keep ordered outcomes distinct when the experiments have labelled first and second stages. A grid shows completeness and prevents duplicate counting; unequal probabilities need weights rather than a simple count.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines sample space?
The complete set of outcomes in an experiment.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
A coin and a fair die have 12 equally likely ordered outcomes: H1 to H6 and T1 to T6. Heads with an even die result has three outcomes, so probability is 3/12=1/4. Two dice have 36 ordered pairs. Total seven arises from (1,6),(2,5),(3,4),(4,3),(5,2),(6,1), so probability is 6/36=1/6. A double has six outcomes and probability 1/6. Total at least eleven occurs in (5,6),(6,5),(6,6), giving 1/12. With a spinner divided into unequal sectors, the named colours are not automatically equally likely; use the sector proportions or a justified experimental estimate.
Systematic possibilities and equally likely outcomes
List outcomes systematically using a grid, table or tree
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
How many ordered outcomes do two fair dice have?
6 choices for the first and 6 for the second: 36.
Test a tempting shortcut
- Counting totals rather than equally likely dice pairs gives wrong weights. State whether order matters. A possibility table lists outcomes; its entries are not automatically equiprobable.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
All possible totals of two fair dice are equally likely. This claim is false. Explain which definition or assumption it violates.
How many ordered pairs total seven?
List (1,6) through (6,1).
All possible totals of two fair dice are equally likely.
Counting totals rather than equally likely dice pairs gives wrong weights. State whether order matters. A possibility table lists outcomes; its entries are not automatically equiprobable.
Interpret a new situation
- AQA P6/P7 uses tables, grids and trees to enumerate theoretical possibilities. Explain why the selected elementary outcomes have equal probability before dividing counts.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find probability of heads and an even result for coin and die.
3 favourable outcomes out of 12.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Foundation · 3.5. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The complete set of outcomes in an experiment. Choose the relationship, show the method, check its assumptions and interpret the result.