サンプリングによる推論、期待値および順列・組合せ
導入済み| English | 意味 |
|---|---|
| expected value/ekˈspektɪd ˈvæljuː/ | 期待値 |
| combination/ˌkɒmbɪˈneɪʃn/ | 組合せ |
A decision before an answer
- Choosing a captain and deputy is different from choosing two unranked representatives. Roles make order matter.
- Your goal: Interpret a population estimate and simulation evidence without claiming proof.
Read the relationship
- A random sample supports inference to its source population; a margin of error gives a sampling-uncertainty interval around an estimate. Random assignment instead supports a causal treatment comparison. In a simulation, a result rarely produced under a claim can be evidence against that claim; failure to find an unusual result does not prove the claim true.
- Calculate conditional, compound and expected-value quantities.
Choose two unranked representatives from four people. Number of choices:
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
4·3/2=6; order does not matter.
Use the defining rule
- Conditional probability restricts the denominator to the conditioning group. For either A or B, subtract their overlap after adding separate probabilities. Independence permits multiplication of unchanged probabilities; without-replacement draws require updated counts. Expected value is the sum of each possible outcome times its probability, not necessarily an outcome achieved in one trial.
- Choose permutations or combinations according to whether order matters.
A random award is 10 with probability 0.2 and 0 otherwise. Expected award:
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
10·0.2+0·0.8=2.
Check the conditions
- Use the multiplication principle for successive choices. For n distinct objects, choosing r in order gives n!/(n-r)!; choosing an unranked set gives n!/[r!(n-r)!]. Dividing by r! removes the multiple orderings of the same set. These formulas assume distinct objects and no replacement; different conditions need a corresponding count.
- Choose permutations or combinations according to whether order matters.
From five students, captain/deputy pairs number 5·4=20; unranked pairs number 5·4/2=10. A game awards 4 points with probability 1/4 and 0 otherwise, so expected award is 1 point per play. A sample estimate 52% ±3 percentage points gives 49%–55%; it is not a guarantee about every sample or individual.
An estimate 64% ±4 points has lower endpoint ____%.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
Subtract four percentage points.
Apply the task format
- Translate the event to a count before calculating its probability. If a favourable event can occur in several disjoint ways, add those ways; if paths overlap, avoid double-counting. A numerical answer must be between zero and one for a probability, while an expected cost or score has the units of the outcomes.
- Choose permutations or combinations according to whether order matters.
Order, replacement and conditioning change the calculation. Failing to reject a statistical claim is not proving it true.
このケースに当てはまる答えはどれか？
検証された証拠であると主張することなく、母集団推計とシミュレーション結果を解釈する
Failing to reject a claim establishes that it is certainly true.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
It means the evidence did not justify rejection under the procedure.
Keep the distinctions
- expected value 期望值 — A probability-weighted average of possible outcomes.
- combination 组合 — An unordered selection of distinct objects.
- Interpret a population estimate and simulation evidence without claiming proof.
- Calculate conditional, compound and expected-value quantities.
- Choose permutations or combinations according to whether order matters.
Match each term with its precise meaning in this lesson.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
難易度 3/5 · 200 コイン + 今日初回完了時に小ボーナス付与
The order follows the stated process; check each stage before the next.