---
# System prepended metadata

title: 符文大地傳說提摩蘑菇模型
tags: [math, Public, LOR]

---

###### tags: `math` `LOR` `Public`

# 符文大地傳說提摩蘑菇模型

## 命題

給定
1. 牌庫數量n
2. 牌庫裡蘑菇總數量m

求此次抽取的牌上共有幾個蘑菇？

### 蘑菇模型說明
1. 蘑菇會隨機種在牌庫的其中一張牌
2. 一張牌上可以有複數個蘑菇



## 解題

### 公式解

$P(n,m,k)=C^m_k\cdot(\frac{1}{n})^k\cdot(\frac{n-1}{n})^{m-k}$  

n=牌庫剩餘值
m=蘑菇總數量
k=抽出蘑菇數量

$P(n,1,1)= \frac {1}{n}$  
$P(n,2,1)=2 \times  \frac {1}{n} \times  \frac {n-1}{n}$  
$P(n,2,2)={( \frac {1}{n})}^2$  
$P(10,20,2)=C^{20}_2 \times ( \frac {1}{10})^2 \times ( \frac {9}{10})^{18}  \approx 0.2852$

### 表列整理
 k | P(10,10,k) | P(10,20,k) | P(10,30,k) | P(10,40,k)
--:|-----------:|-----------:|-----------:|----------:
 0 |     34.86% |     12.15% |      4.23% |      1.47%
 1 |     38.74% |     27.01% |     14.13% |      6.56%
 2 |     19.37% |     28.51% |     22.76% |     14.23%
 3 |      5.73% |     19.01% |     23.60% |     20.03%
 4 |      1.11% |      8.97% |     17.70% |     20.58%
 5 |      0.14% |      3.19% |     10.23% |     16.47%
 6 |      0.01% |      0.88% |      4.73% |     10.67%
 7 |      0.00% |      0.19% |      1.80% |      5.76%
 8 |      0.00% |      0.03% |      0.57% |      2.64%
 9 |      0.00% |      0.00% |      0.15% |      1.04%
10 |      0.00% |      0.00% |      0.03% |      0.35%
11 |      0.00% |      0.00% |      0.00% |      0.10%
12 |      0.00% |      0.00% |      0.00% |      0.02%
13 |      0.00% |      0.00% |      0.00% |      0.00%
14 |      0.00% |      0.00% |      0.00% |      0.00%
15 |      0.00% |      0.00% |      0.00% |      0.00%

> 小心地上有蘑菇~