---
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).