Systematic possibilities and equally likely outcomes · Higher
導入済み| English | 意味 |
|---|---|
| sample space/ˈsæmpl speɪs/ | 標本空間 |
The possible totals are not equally likely.
- Two fair dice can total seven in six different ways, but total two in only one way.
- This lesson studies sample space 样本空间: The complete set of outcomes in an experiment.
The method
- 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.
Why are the totals of two dice NOT equally likely, even though the 36 pairs are?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
Equally likely units are the ordered PAIRS. A total collects several pairs - 7 comes from six of them, giving 6/36=1/6 - so the totals inherit unequal weights.
A worked case
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?
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
6 choices for the first and 6 for the second: 36.
Where it goes wrong
- 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.
Not true: All possible totals of two fair dice are equally likely. A total of 7 has six ways, but 2 has only one, so the totals are not equally likely.
How many ordered pairs total seven?
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
List (1,6) through (6,1).
All possible totals of two fair dice are equally likely.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
A total of 7 has six ways, but 2 has only one, so the totals are not equally likely.
Applying it
- AQA P6/P7 uses tables, grids and trees to enumerate theoretical possibilities.
- Explain why the selected elementary outcomes have equal probability before dividing counts.
Find probability of heads and an even result for coin and die.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
3 favourable outcomes out of 12.
Match each two-dice event to its probability.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Count the ordered pairs in the grid that satisfy the event; the grid keeps (4,3) and (3,4) distinct.
In your course
- AQA GCSE Mathematics (8300), Higher tier, specification section 3.5.
- Paper 1 is non-calculator: keep fractions, surds and π exact and show each step. On calculator papers, still write the expression before its value.
- Sample space: The complete set of outcomes in an experiment.
- List outcomes systematically using a grid, table or tree.
- Watch for: counting totals rather than equally likely dice pairs gives wrong weights.