hermes 初始配置

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+---
+name: dice-wildcard-probability
+category: research
+description: Calculate probability that, when rolling n dice (standard six-sided) with a designated wildcard face that can count as any other face, at least k dice show the same face.
+---
+
+## Description
+Calculate the probability that, when rolling n dice (standard six-sided) with a designated wildcard face (e.g., 1) that can count as any other face, at least k dice show the same face (using wildcards as needed).
+
+## When to Use
+- Analyzing dice games with wildcards.
+- Probability homework or game design.
+
+## Assumptions
+- Dice are fair and independent.
+- Wildcard face value is known (default: 1).
+- Other faces are 2..6 (5 regular faces).
+- We want probability of ≥ k matches for any face (including using wildcards).
+
+## Method Overview
+1. Let n = total dice, p_w = probability of wildcard = 1/6, p_r = 5/6 for regular.
+2. Let w = number of wildcards observed; w ~ Binomial(n, p_w).
+3. Given w wildcards, remaining r = n - w dice are regular, uniformly distributed over 5 faces.
+4. Need probability that among r regular dice, some face appears at least t = k - w times (if t ≤ 0, condition already satisfied; if t > r, impossible).
+5. For regular dice, compute P_cond(t, r) = Prob{max count ≥ t} using inclusion–exclusion over subsets of faces:
+   \[
+   P_{\text{cond}}(t,r)=\sum_{s=1}^{5}(-1)^{s+1}\binom{5}{s}
+   \sum_{y=st}^{r}\binom{r}{y}\left(\frac{s}{5}\right)^{y}
+   \left(1-\frac{s}{5}\right)^{r-y}
+   \frac{\displaystyle\binom{y-st+s-1}{s-1}}{s^{y}}.
+   \]
+6. Overall probability:
+   \[
+   P(\ge k)=\sum_{w=0}^{n}\binom{n}{w}p_w^{w}(1-p_w)^{n-w}
+   \times
+   \begin{cases}
+   1 & t\le 0\\
+   0 & t> r\\
+   P_{\text{cond}}(t,r) & \text{otherwise}
+   \end{cases}
+   \]
+   where \(t = k - w\) and \(r = n - w\).
+
+## Steps (for n=10, wildcard=1)
+1. Precompute binomial probabilities for w=0..10.
+2. For each w, compute t = k - w, r = 10 - w.
+3. If t ≤ 0 → contribution = binom prob.
+4. Else if t > r → contribution = 0.
+5. Else compute P_cond(t,r) using the formula (can be implemented via a short script).
+6. Sum contributions.
+
+## Example Results (n=10)
+| k (≥ matches) | Probability |
+|---------------|-------------|
+| 3 | 0.9981245713 |
+| 4 | 0.9112991700 |
+| 5 | 0.6298112188 |
+| 6 | 0.2928175250 |
+| 7 | 0.0877220349 |
+| 8 | 0.0163763622 |
+| 9 | 0.0017599261 |
+|10 | 0.0000846093 |
+
+## Implementation Tips
+- Use logarithms for large factorials if needed.
+- The inner sum over y can be limited to y from s*t to r.
+- For small n (≤20) direct computation is fine.
+- Verify with Monte Carlo simulation for sanity.
+
+## Pitfalls
+- Forgetting that wildcard can be used for any face, not just a specific one.
+- Miscalculating the inclusion‑exclusion sign.
+- Overlooking the case t ≤ 0 (already satisfied).
+
+## References
+- Derived using generating functions and inclusion–exclusion.
+- See also: “Probability of dice matches with wildcards” (standard technique).
+
+## How to Extend
+- Change number of regular faces (e.g., different dice).
+- Change wildcard probability (e.g., multiple wildcard faces).
+- Compute probability of exactly k matches by subtracting P(≥k+1) from P(≥k).