Use Python to Emulate Sampling with Replacement

罐子裡有100顆球，紅色的球佔40%，綠色的球佔60%。從罐子裡取出10顆球，希望出現最多1顆紅色球的機率是多少？

這是一個Sampling with Replacement(放回取樣)問題，在Peak Balls from a Bin，我們用簡單的機率公式，可以算出來：

P [# = 0] = 0.6^10 = 0.006

P [# = 1] = 0.6^9 * 0.4 * 10

P [v <= 0.1] = P [# = 0] + P [# = 1] = 0.046…

本篇，我用Python給定不同的母體數量，100、1000、10000、100000，模擬這個Sampling with Replacement。

程式碼如下：

[SamplingWithReplacement.py]

[Result]

執行結果如下：

There are 100 balls in the bin. (40 red balls, 60 green balls)

Select %d balls from the bin with replacement.

Times of replacement               = 10000

Number of red balls in the sample  = 348

Probability                        = 0.034800

There are 1000 balls in the bin. (400 red balls, 600 green balls)

Select %d balls from the bin with replacement.

Times of replacement               = 10000

Number of red balls in the sample  = 469

Probability                        = 0.046900

There are 10000 balls in the bin. (4000 red balls, 6000 green balls)

Select %d balls from the bin with replacement.

Times of replacement               = 10000

Number of red balls in the sample  = 473

Probability                        = 0.047300

There are 100000 balls in the bin. (40000 red balls, 60000 green balls)

Select %d balls from the bin with replacement.

Times of replacement               = 10000

Number of red balls in the sample  = 438
Probability                        = 0.043800

我們發現，母體數量愈大，Probability趨近於一個極限值，大概在0.046附近。

-Count

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