一只口袋装有形状、大小完全相同的3只小球,其中红球、黄球、黑球各1只.现从口袋中先后有放回地取球
次
,且每次取1只球,
表示
次取球中取到红球的次数,
,则
的数学期望为______ (用
表示).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2d51f9147b8265c0276c1f2c2659197.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff44a0465a828ecf1a3c0f0483e32322.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2d51f9147b8265c0276c1f2c2659197.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a829fdd8ec0f3b7ede883cf2c3e53b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
2024·湖北武汉·模拟预测 查看更多[4]
更新时间:2024-06-08 12:19:05
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【推荐1】考查等式:
(*),其中
,
且
.某同学用概率论方法证明等式(*)如下:设一批产品共有
件,其中
件是次品,其余为正品.现从中随机取出
件产品,记事件
{取到的
件产品中恰有
件次品},则
,
,1,2,…,
.显然
,
,…,
为互斥事件,且
(必然事件),因此
,所以
,即等式(*)成立.对此,有的同学认为上述证明是正确的,体现了偶然性与必然性的统一;但有的同学对上述证明方法的科学性与严谨性提出质疑.现有以下四个判断:①等式(*)成立,②等式(*)不成立,③证明正确,④证明不正确,试写出所有正确判断的序号___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb037e045b5418574fe43786d011b870.png)
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【推荐2】“杨辉三角”(或“贾宪三角”),西方又称为“帕斯卡三角”,实际上帕斯卡发现该规律比贾宪晚500多年,若将杨辉三角中的每一个数
都换成分数
,就得到一个如图所示的分数三角形数阵,被称为莱布尼茨三角形.从莱布尼茨三角形可以看出
,其中![](https://staticzujuan.xkw.com/quesimg/Upload/formula/695797cbadfe77a292710550f4482427.png)
________ (用r表示);令
,则
的值为________ .
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【推荐1】若
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1196db72f1d2df1ded968a6762c37bab.png)
___________ .
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【推荐2】已知函数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c81c5965300812a7c849845586d9d53.png)
,
,其中
,
,
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/723383b2c4473b1737c7ecc5ea926625.png)
______ ,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7e054ca12607b76baf8670664b31889.png)
______ .
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【推荐3】设
为
的展开式的各项系数之和,
,
,
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的最小值为__________ .
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【推荐1】甲、乙两名运动员进行乒乓球比赛,规定每局比赛胜者得1分,负者得0分,比赛一直进行到一方比另一方多两分为止,多得两分的一方赢得比赛.已知每局比赛中,甲获胜的概率为
,乙获胜的概率为
,
,且每局比赛结果相互独立.
①若
,则甲运动员恰好在第4局比赛后赢得比赛的概率为____________ ;
②若比赛最多进行5局,则比赛结束时比赛局数
的期望
的最大值为____________ .
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①若
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②若比赛最多进行5局,则比赛结束时比赛局数
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【推荐2】将字母a,a,b,b,c,c放入3×2的表格中,每个格子各放一个字母,则每一行的字母互不相同,且每一列的字母也互不相同的概率为______ ;若共有k行字母相同,则得k分,则所得分数
的均值为______ .
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【推荐1】某机器有四种核心部件A,B,C,D,四个部件至少有三个正常工作时,机器才能正常运行,四个核心部件能够正常工作的概率满足为
,
,且各部件是否正常工作相互独立,已知
,设
为在
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【推荐2】袋中装有5个相同的红球和2个相同的黑球,每次从中抽出1个球,抽取3次按不放回 抽取,得到红球个数记为X,得到黑球的个数记为Y;按放回 抽取,得到红球的个数记为
.下列结论中正确的是________ .
①
;②
;③
;④
.
(注:随机变量X的期望记为
、方差记为
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b734e8f1546481e3eb4976008a045de.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bf0bad68e96b0dfdbe9c461c60c9a31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bac2f9a7b977316691620148a0f3983.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f11dc29a2fe6cd9d8aefceb8493c703a.png)
(注:随机变量X的期望记为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bf3baba074e8aeb6f3ea117865bbd1b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90a0722562d03a0a55a6c63e5d4cc338.png)
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【推荐3】切比雪夫不等式是19世纪俄国数学家切比雪夫(1821.5~1894.12)在研究统计规律时发现的,其内容是:对于任一随机变量
,若其数学期望
和方差
均存在,则对任意正实数
,有
.根据该不等式可以对事件
的概率作出估计.在数字通信中,信号是由数字“0”和“1”组成的序列,现连续发射信号
次,每次发射信号“0”和“1”是等可能的.记发射信号“1”的次数为随机变量
,为了至少有
的把握使发射信号“1”的频率在区间
内,估计信号发射次数
的值至少为______ .
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