Count 朱: 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]

	#
	# Use random numbers to emulates the Sampling with Replacement.
	#
	# We can specify the number of balls in the bin with 40% red balls and 60% green balls.
	# We select 10 balls from the bin and calculate the probability of the red
	# balls in the samples.
	#

	import random

	def sampleingWithReplacement (total, replaceTimes):

	balls = []

	red = int (total * 40 / 100)
	green = total - red
	for i in range (red):
	balls.append (1)

	for i in range (green):
	balls.append (0)

	#print ("Display balls in the bin.")
	#print (balls)

	numberOfRed = 0
	for i in range (replaceTimes):

	random.shuffle (balls)
	sample = balls [0: 10]
	s = sum (sample)
	if s == 0 or s == 1:
	numberOfRed += 1

	p = numberOfRed / replaceTimes
	print ("There are %d balls in the bin. (%d red balls, %d green balls)" % (total, red, green))
	print ("Select %d balls from the bin with replacement.")
	print ("Times of replacement = %d" % replaceTimes)
	print ("Number of red balls in the sample = %d" % numberOfRed)
	print ("Probability = %f" % p)
	print ("")

	if __name__ == '__main__':

	replaceTimes = 10000
	sampleingWithReplacement (100, replaceTimes) # total = 100, probability = 0.034800
	sampleingWithReplacement (1000, replaceTimes) # total = 1000, probability = 0.046900
	sampleingWithReplacement (10000, replaceTimes) # total = 10000, probability = 0.047300
	sampleingWithReplacement (100000, replaceTimes) # total = 100000, probability = 0.043800

[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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Count 朱