administrator
82 lines
3.1 KiB
Markdown
82 lines
3.1 KiB
Markdown
---
|
||
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). |