一小虫在正四面体表面从一顶点到另一顶点概率相同.现掷骰子决定是否移动.若掷出偶数,小虫不动,奇数,小虫移动.若小虫到达B点,游戏结束.最多投5次,求投掷次数的期望

Question

一小虫在正四面体表面从一顶点到另一顶点概率相同.现掷骰子决定是否移动.若掷出偶数,小虫不动,奇数,小虫移动.若小虫到达B点,游戏结束.最多投5次,求投掷次数的期望

Answer 1

p(?=1)=(1/2)*(1/3)=1/6
p(?=2)=(1/2)^2*(1/3+(1/3)*(2/3))=5/36
p(?=3)=(1/2)^3*(1/3+(1/3)*2*(2/3)+(1/3)*(2/3)^2)=25/216
p(?=4)=(1/2)^4*(1/3+(1/3)*3*(2/3)+(1/3)*3*(2/3)^2+(1/3)*(2/3)^3)=125/1296
p(?=5)=(1/2)^5*((1/3+(1/3)*4*(2/3)+(1/3)*6*(2/3)^2+(1/3)*4*(2/3)^3+(1/3)*(2/3)^4)+(1+5*(2/3)+10*(2/3)^2+10*(2/3)^3+5*(2/3)^4+(2/3)^5))=625/1296
E(?)=(1/6)*1+(5/36)*2+(25/216)*3+(125/1296)*4+(625/1296)*5=4651/1296