config-hermes/skills/research/dice-wildcard-probability/SKILL.md

name, category, description

name	category	description
dice-wildcard-probability	research	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).