Question

一个工厂生产的产品中废品率为0.005,任取1000件,计算其中不超过5件废品的概率

Answer 1

都是正品的概率=(1-0.005)^1000=0.006654,
恰有1件废品的概率=C(1000，1)*0.005*0.995^999=0.033437,
恰有2件废品的概率=C(1000，2)*0.005^2*0.995^998=0.083928,
恰有3件废品的概率=C(1000，3)*0.005^3*0.995^997=0.140303,
恰有4件废品的概率=C(1000，4)*0.005^4*0.995^996=0.175731,
恰有5件废品的概率=C(1000，5)*0.005^5*0.995^995=0.175908.
把上述6个概率相加就得所求（计算略）.

Answer 2

设1000件产品中废品有X件，则X服从二项分布N(1000,0.005).
EX=1000*0.005=5，DX=1000*0.005*(1-0.005)=4.975
由于n=1000,故X近似服从正态分布N(5,4.975),则
P{X≤5}=P{(X-5)/4.975≤0}=Φ(0)=0.5

Answer 3

p=C(1000,0)*(1-0.005)^1000+C(1000,1)*0.005^1*(1-0.005)^999
+C(1000,2)*0.005^2*(1-0.005)^998+C(1000,3)*0.005^3*(1-0.005)^997
+C(1000,4)*0.005^4*(1-0.005)^996+C(1000,5)*0.005^5*(1-0.005)^995
=0.0066539685788319948734572146875982
+0.033437028034331632529935752198986
+0.083928620618837439440718131775846
+0.1403027862271352840231770444092
+0.17573100234730386704913004180399
+0.17590761642001472520696836345404
=0.616