掷n面色子,每掷1次都把得到的数字记下来,掷到他们的和能被n整除则停.求平均要掷多少次原题 A fair n-sided die is rolled repeatedly and the sum of the rolls is recorded.The process stops as soon asthe recorded sum is d
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掷n面色子,每掷1次都把得到的数字记下来,掷到他们的和能被n整除则停.求平均要掷多少次原题 A fair n-sided die is rolled repeatedly and the sum of the rolls is recorded.The process stops as soon asthe recorded sum is d
掷n面色子,每掷1次都把得到的数字记下来,掷到他们的和能被n整除则停.求平均要掷多少次
原题
A fair n-sided die is rolled repeatedly and the sum of the rolls is recorded.The process stops as soon as
the recorded sum is divisible by n.What is the expected number of rolls when the process stops?3q
掷n面色子,每掷1次都把得到的数字记下来,掷到他们的和能被n整除则停.求平均要掷多少次原题 A fair n-sided die is rolled repeatedly and the sum of the rolls is recorded.The process stops as soon asthe recorded sum is d
首先抛色子应该是均匀分布
所以每个面的概率都是1/n
抛一次得出的点数的数学期望EX=Σi*1/n i=1→n
EX=1/n·Σi i=1→n=1/n·n（n+1）/2=(n+1)/2
抛j次点数的和的数学期望
ΣEX i=1→j = j·(n+1)/2
由题意有 ΣEX i=1→j =n· k k∈N
于是 j·(n+1)/2= n· k
j=[n/(n+1)]·2k k∈N
由题意知 n≥2 于是
当n=2p时 k=2p+1 k∈N p∈N
j=2n
当n=2p+1时 k=p+1 k∈N p∈N
j=n
于是 当n为奇数 抛n次 当n为偶数 抛2n次