###### 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~