In the book "[The Mathematical Experience](http://books.google.com/books?id=lMdz84dWNnAC&q=Doob#v=snippet&q=Doob&f=falseBlockquoteblah)" it says: 



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> "An infinite [binary] sequence $x\_1, x\_2, \ldots$ is called random in the sense of von Mises if every infinite sequence $x\_{n\_1}, x\_{n\_2}, \ldots$ extracted from it and determined by a policy or rule R is $\infty$-distributed. Now comes the shocker. It has been established by Joseph Doob that there are no sequences that are random in the sense of von Mises." 
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A sequence on $\{H,T\}$ is $\infty$-distributed if for each positive integer $k$ and sequence $\vec y \in \{H,T\}^k$ the set $\{n\in {\mathbb N} \colon \langle x\_{n},\dots,x\_{n+k-1}\rangle=\vec y\}$ has density $2^{-k}$.


But the definition of von Mises seems so natural to me that if a sequence does not satisfy it then the sequence is not random. 



Did Joseph Doob prove that random sequences don't exist?
At Gerald Edgar's suggestion, I promote my comment to an answer. 


There is a good discussion of the questions raised here in the chapter on randomness in Seminumerical Algorithms, Volume 2 of Knuth's The Art Of Computer Programming.


