一个均匀的硬币投掷10次,令i/j为正面不连着出现的概率,其中i,j互质,求i+j

Question

一个均匀的硬币投掷10次,令i/j为正面不连着出现的概率,其中i,j互质,求i+j

Answer 1

假设P(X)为“正面共出现X次且正面不连着出现的概率”,
P(0)=(1/2)^10=1/1024
P(1)=C(10,1)*(1/2)^10=10/1024
P(2)=(C(10,2)-C(9,1))*(1/2)^10=36/1024
P(3)=(C(7,3)+C(7,2))*(1/2)^10=561024
P(4)=(C(6,4)+C(6,3))*(1/2)^10=35/1024
P(5)=2*(1/2)^10=2/1024
===> P(正面不连着出现)=P(0)+…+P(5)=140/1024=35/256
===> i+j=291