解题方法
1 . 现有甲、乙两个盒子中都有大小、形状、质地相同的2个红球和1个黑球.从两个盒子中各任取一个球交换,记为一次操作.重复进行
次操作后,记甲盒子中黑球个数为
,甲盒中恰有1个黑球的概率为
,恰有2个黑球的概率为
.
(1)求随机变量
的分布列;
(2)求数列
的通项公式;
(3)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6368fec0c2c25db7c29b014d60270e97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93d0f3799612b81e85b87241ec8eee68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
(1)求随机变量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/096b1ece1dcd29c59a46a4b3e02cb548.png)
(2)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e067aeb203bc5ce00c9dc47aa34cee5e.png)
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2 . 用n个不同的元素组成m个非空集合(
,每个元素只能使用一次),不同的组成方案数记作
例如,用1,2,3,4这4个元素组成2个非空集合共有7种方案,即
;
;
;
;
;
;
.于是
.
(1)求和:
;
(2)证明:当
时,
;
(3)某系列手办盲盒共装有4种不同款式的手办,打开其中任何一个盲盒都可以获得1个手办(款式随机,且获得每种款式的概率都相同)
①求购买该系列盲盒7盒就能集齐全部4种款式的概率p;
②设购买该系列盲盒7盒能获得不同手办款式的种类数为随机变量X,求X的数学期望
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/315109103349a6e41373c994e89f9f51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca6d3696e6ff2775f4cde52a66a3dc53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1cba028fe4a1f51166cf2492a5002183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c0a20cc4a02e6c8f4dfc71f3fb58b8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77b29beb494332847849d4725ea91ea6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b40289da7846a67d8905322ad257de95.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a4a62a63babd750bc283099834a2373.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebd2bbf4e475c3a9ed086cc6906cb87e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3957c2adc4e5f497ecc9d06c64c96f73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1edc67d4416e549c2d01a82ac2f370d4.png)
(1)求和:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36901a78636c643523467783239274e2.png)
(2)证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33ed19a52f57f9edb69e52e3d7afef49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20f4077061e91eea5fbc73ab9aae7a54.png)
(3)某系列手办盲盒共装有4种不同款式的手办,打开其中任何一个盲盒都可以获得1个手办(款式随机,且获得每种款式的概率都相同)
①求购买该系列盲盒7盒就能集齐全部4种款式的概率p;
②设购买该系列盲盒7盒能获得不同手办款式的种类数为随机变量X,求X的数学期望
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bf3baba074e8aeb6f3ea117865bbd1b.png)
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解题方法
3 . 某汽车厂商生产某型号具有自动驾驶功能的汽车,该型号汽车配备两个相互独立的自动驾驶系统(记为系统
和系统
),该型号汽车启动自动驾驶功能后,先启动这两个自动驾驶系统中的一个,若一个出现故障则自动切换到另一个系统.为了确定先启动哪一个系统,进行如下试验:每一轮对系统
和
分别进行测试试验,一轮的测试结果得出后,再安排下一轮试验.当一个系统出现故障的次数比另一个系统少2次时,就停止试验,并认为出现故障少的系统比另一个系统更稳定.为了方便描述问题,约定:对于每轮试验,若系统
不出现故障且系统
出现故障,则系统
得1分,系统
得-1分;若系统
出现故障且系统
不出现故障,则系统
得-1分,系统
得1分;若两个系统都不出现故障或都出现故障,则两个系统均得0分.系统
出现故障的概率分别记为
和
,一轮试验中系统
的得分为
分.
(1)求
的分布列;
(2)若系统
和
在试验开始时都赋予2分,
表示“系统
的累计得分为
时,最终认为系统
比系统
更稳定”的概率,则
,
,其中
.现根据
的值来决定该型号汽车启动自动驾驶功能后先启动哪个系统,若
,则先启动系统
;若
,则先启动系统
;若
,则随机启动两个系统中的一个,且先启动系统
的概率为
.
①证明:
;
②若
,由①可求得
,求该型号汽车启动自动驾驶功能后无需自动切换到另一个自动驾驶系统的概率.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20ebaa32f4f1f4f807ca9aeb7fb29951.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(2)若系统
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82078c39ec148befeb0011cc261a51c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e86f01de277cfcf6c293f35f5fdf2bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c59c570f0c7b0702dfdb16088da7ec86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/246e3c7846e0d306ed9006b77e7b7697.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adad9633b73dfbbb3d84b4f15979e99e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25444b9b87ffa695464502f5e8864fb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9e5dc72fd2623e4766994baf3fbabc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85ef764dca7fcf36caa9395387ddf4f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adad9633b73dfbbb3d84b4f15979e99e.png)
①证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c75d40514ea42bc2a92a25840a0b2251.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97df32a5e8c500aca06215fe46836999.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91691dece5b2d47cf1fcc0797c817548.png)
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2024-05-23更新
|
602次组卷
|
3卷引用:福建省漳州市2024届高三毕业班第四次教学质量检测数学试题
名校
4 . 情报
是仅含0和1两种的k位数据,例如11001. 情报传输时要经过n个信号站,每经过一个信号站,每位数字0传错为1的概率为
,每位数字1传错为0的概率为
,其中
,在各次传输过程中,情报中各数字相互独立,且传输中无其他错误发生. 情报
经过n个信号站传输后的情报为
,设
与
完全相同的概率为
,
与
中有
个对应位置数字取值相等.
(1)若
,
,求
的分布列;
(2)若
,证明
的数学期望
与n无关;
(3)若
,且
,证明:
. 若将
改为
,判断是否仍有
恒成立,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fb44db1dc864ff4901be1e10da79747.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8be646cd52d7f2f1714e7542e75810f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adad9633b73dfbbb3d84b4f15979e99e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/149e9da3d622abc5e4ef8e84f37e37bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fb44db1dc864ff4901be1e10da79747.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ddad3d9fdb5e9951b6a1c31f9a72a71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ddad3d9fdb5e9951b6a1c31f9a72a71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fb44db1dc864ff4901be1e10da79747.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ddad3d9fdb5e9951b6a1c31f9a72a71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fb44db1dc864ff4901be1e10da79747.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d9af571e201cdcc0279d8e0e73aba1c.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42e988a40e6b2ea823aeb61c8e90d806.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/933ed5f7233739d6d3e02756560b0d85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5031a3a951c4a1d1c5e9f80a5e26513.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1bcf0584660fadc47dda5241aef29b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93d0f3799612b81e85b87241ec8eee68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/229494e1240a594592035d23283fedbc.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9eea4189f53fad56c7d690cec48c7856.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7fc0e595dbe11436b569efea1a4f40a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fc6eaa8c959dee14c16e8d3fbdc9088.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9eea4189f53fad56c7d690cec48c7856.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55eaeed009493bb63365376920c97372.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fc6eaa8c959dee14c16e8d3fbdc9088.png)
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2024·全国·模拟预测
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5 . 甲、乙两人进行象棋比赛,赛前每人有3面小红旗.一局比赛后输者需给赢者一面小红旗;若是平局不需要给红旗,当其中一方无小红旗时,比赛结束,有6面小红旗者最终获胜.根据以往的两人比赛结果可知,在一局比赛中甲胜的概率为0.5,乙胜的概率为0.4.
(1)若第一局比赛后甲的红旗个数为X,求X的分布列和数学期望;
(2)若比赛一共进行五局,求第一局是乙胜的条件下,甲最终获胜的概率(结果保留两位有效数字);
(3)记甲获得红旗为
面时最终甲获胜的概率为
,
,
,证明:
为等比数列.
(1)若第一局比赛后甲的红旗个数为X,求X的分布列和数学期望;
(2)若比赛一共进行五局,求第一局是乙胜的条件下,甲最终获胜的概率(结果保留两位有效数字);
(3)记甲获得红旗为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e7435d45cd9df9a16bc01188c8fdef1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e94b1e988f6574093ecf0675049af801.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0644cc6e89583bcb9564d85a80ee6c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92b0e645eb76eaea9a16d406e85f2cad.png)
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6 . 甲、乙两名小朋友,每人手中各有3张龙年纪念卡片,其中甲手中的3张卡片为1张金色和2张银色,乙手中的3张卡片都是金色的,现在两人各从自己的卡片中随机取1张,去与对方交换,重复
次这样的操作,记甲手中银色纪念卡片
张,恰有2张银色纪念卡片的概率为
,恰有1张银色纪念卡片的概率为
.
(1)求
的值.
(2)问操作几次甲手中银色纪念卡片就可能首次出现0张,求首次出现这种情况的概率
.
(3)记
.
(i)证明数列
为等比数列,并求出
的通项公式.
(ii)求
的分布列及数学期望.(用
表示)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ffb021aa7d5a5c2f0691e337caad624.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ce603aa3abcb61750d2191aaa13dddc.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7f66b7e38f44f8cd5d48b3aa24a20fc.png)
(2)问操作几次甲手中银色纪念卡片就可能首次出现0张,求首次出现这种情况的概率
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
(3)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9131abf93295537bbc0c54a8c42e88e2.png)
(i)证明数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c895d4ce5ce82ef9b311b9369b4de11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(ii)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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解题方法
7 . 在三维空间中,单位立方体的顶点坐标可用三维坐标
表示,其中
.而在
维空间中
,以单位立方体的顶点坐标可表示为
维坐标
,其中
.现有如下定义:在
维空间中,
,
两点的曼哈顿距离为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd16a6bce68922e270867b93251aa45b.png)
(1)在3维单位立方体中任取两个不同顶点,试求所取两点的曼哈顿距离为1的概率;
(2)在
维单位立方体中任取两个不同顶点,记随机变量
为所取两点间的曼哈顿距离
(i)求出
的分布列与期望;
(ii)证明:随机变量
的方差小于
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4c2a29087dbd2e7635da13f7d288c1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/489ac1b21a2d8f1cf07dc4aaca39a2bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/677fd74842cbce34aed7073cebbd9c58.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3183165f26bf33a2f3e7b6354937524e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/778f11ad6b2139da11259859c06e868c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f173dd2b7c86ba2ec88c614ad334bb37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdd16f86170db5efa19732f33aad277b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd16a6bce68922e270867b93251aa45b.png)
(1)在3维单位立方体中任取两个不同顶点,试求所取两点的曼哈顿距离为1的概率;
(2)在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4f27f84764f1cca89ce3d93fc1cf603.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(i)求出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(ii)证明:随机变量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fe33278e4c69efacf81defce3045cec.png)
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名校
8 . 第二次世界大战期间,了解德军坦克的生产能力对盟军具有非常重要的战略意义.已知德军的每辆坦克上都有一个按生产顺序从1开始的连续编号.假设德军某月生产的坦克总数为N,随机缴获该月生产的n辆(
)坦克的编号为
,
,…,
,记
,即缴获坦克中的最大编号.现考虑用概率统计的方法利用缴获的坦克编号信息估计总数N.
甲同学根据样本均值估计总体均值的思想,用
估计总体的均值,因此
,得
,故可用
作为N的估计.
乙同学对此提出异议,认为这种方法可能出现
的无意义结果.例如,当
,
时,若
,
,
,则
,此时
.
(1)当
,
时,求条件概率
;
(2)为了避免甲同学方法的缺点,乙同学提出直接用M作为N的估计值.当
,
时,求随机变量M的分布列和均值
;
(3)丙同学认为估计值的均值应稳定于实际值,但直观上可以发现
与N存在明确的大小关系,因此乙同学的方法也存在缺陷.请判断
与N的大小关系,并给出证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a5e1bb2637455d05313a112c5d745bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/096b1ece1dcd29c59a46a4b3e02cb548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5031a3a951c4a1d1c5e9f80a5e26513.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93d0f3799612b81e85b87241ec8eee68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dcf3400c1490071b390aaac0ad0e102.png)
甲同学根据样本均值估计总体均值的思想,用
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afb1694f46c040a6c976b2ef3eb3934b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/120e4da3fe22be28b3bb28f28fbcc862.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92da09d5877d3dfe1a856b6353b81906.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b7171e0c9c26b9f39a32d3a61d113cf.png)
乙同学对此提出异议,认为这种方法可能出现
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8a1bd336033c63bc9c4f99ff2b482b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebcc133d5b11b33a904875182d8c8261.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be604061cf1591f7069472269d4c9719.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c50a1cf3b1a6f9a12605cbdf48e5de5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49bf4a59874878184dadeec74d1781d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53722e8f43d44f9c611398ddaab151f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afac93e0089a7ffca9a1f720e13b6878.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de3874d2e8c49308151837161d7aa91c.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebcc133d5b11b33a904875182d8c8261.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be604061cf1591f7069472269d4c9719.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b9b7c101f267bbf233da7d3ac30e6f0.png)
(2)为了避免甲同学方法的缺点,乙同学提出直接用M作为N的估计值.当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/270e5f2895909d5b6b6c612a8696565b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fac3649308b528fd56545ba102dc42d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/348388b2590255369527f86fd6be63c3.png)
(3)丙同学认为估计值的均值应稳定于实际值,但直观上可以发现
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/348388b2590255369527f86fd6be63c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/348388b2590255369527f86fd6be63c3.png)
您最近一年使用:0次
2024-06-11更新
|
722次组卷
|
3卷引用:浙江省(杭州二中、绍兴一中、温州中学、金华一中、衢州二中)五校联考2024届高考数学模拟卷
名校
9 . 某校开展科普知识团队接力闯关活动,该活动共有两关,每个团队由
位成员组成,成员按预先安排的顺序依次上场,具体规则如下:若某成员第一关闯关成功,则该成员继续闯第二关,否则该成员结束闯关并由下一位成员接力去闯第一关;若某成员第二关闯关成功,则该团队接力闯关活动结束,否则该成员结束闯关并由下一位成员接力去闯第二关;当第二关闯关成功或所有成员全部上场参加了闯关,该团队接力闯关活动结束.
已知A团队每位成员闯过第一关和第二关的概率均为
,且每位成员闯关是否成功互不影响,每关结果也互不影响.
(1)用随机变量X表示A团队第
位成员的闯关数,求X的分布列;
(2)已知A团队第
位成员上场并闯过第二关,求恰好是第3位成员闯过第一关的概率;
(3)记随机变量
表示A团队第
位成员上场并结束闯关活动,证明
单调递增,并求使
的n的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfc321599521a98661ed719cc82ca87c.png)
已知A团队每位成员闯过第一关和第二关的概率均为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3cfa1e7ffae662aefb49a44c52d4954d.png)
(1)用随机变量X表示A团队第
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a2c1905706959a99c1962ce75e95d8f.png)
(2)已知A团队第
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec6b15d9089613091e9b5f6108f3d180.png)
(3)记随机变量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29c4cbb3a50014fa18fab2e0de87ee22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e75e8665f4987edbf6a62590564235b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab9310b9dc777d33abe9c873a0c729b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38ab9b42bb564a497bdd5cf14d03d8ca.png)
您最近一年使用:0次
2024-04-10更新
|
654次组卷
|
2卷引用:湖南省长沙市第一中学2023-2024学年高二下学期第一次阶段性检测数学试题
2024·全国·模拟预测
10 . 近年来,“盲盒”在许多年轻人中开始流行.小红最近喜欢上了一款单价1元的盲盒.已知盲盒全套共有
款玩偶,小红喜欢其中的一款玩偶.已知小红手里有
元零花钱(
,
),小红每次开一个盲盒,若开出自己喜欢的玩偶则停止,否则再开一个盲盒,直到开出自己喜欢的玩偶或者花完零花钱为止.设小红停止开盲盒时剩余零花钱为
,小红每次开盲盒的结果互不影响.
(1)若
,
,求
的分布列和数学期望;
(2)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3c76b215227be58c281940c091d07f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e98d7d3c68e05f53584eba2e781b8de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be604061cf1591f7069472269d4c9719.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab839d8569171afab5ed55c22013aa72.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a68233ed4d61814f6cb2bee2d2fbe1e1.png)
您最近一年使用:0次