What reason is there to conjecture that every finite string is really in the decimal expansion of $\pi$? One of my students asked me this, and it occurred to me that I had never really questioned it.
Apparently, it is only conjectured but widely believed that the decimal expansion in base $10$ of $\pi$ contains all finite strings of the numerals $0$ through $9$.
Am I even accurate that the conjecture is widely accepted? Regardless, what is the rationale for this belief? Do any good heuristics exist? It seems perfectly logical (dare I say likely) that, just maybe, the string 2347529384759748975847523462346435664900060906, for example, never occurs. The conjecture seems absurdly strong, to me.
And just because a separate question would be ridiculous: does this conjecture extend to other famous transcendental numbers? Is it indeed conjectured that this is a property of transcendental numbers in general?
 A: $\pi$ is generally (though weakly) believed to be normal; that is, its infinite sequence of digits, in any base $b$, contain all $1$-digit sequences with density $1/b$, all $2$-digit sequences with density $1/b^2$, all $3$-digit sequences with density $1/b^3$, and so forth.  In particular, this is believed to be true of $\pi$ in base $10$, for somewhat general reasons:


*

*All real numbers are normal, except for a set that is of Lebesgue measure zero.  Roughly speaking, real numbers are normal with probability $1$.  However, it is in general challenging to prove that a given number is normal (other than those numbers "designed" to be normal), and $\pi$ is no exception.

*Empirical studies on the digit sequence of $\pi$ (in base $10$, as far as I know) have revealed no statistical tendencies that violate normality.
If $\pi$ is normal, then the weaker assertion that it contains any finite digit sequence at least once (let alone contains it as frequently as any other digit sequence of the same length) is true as a consequence.  As far as I know, there is no intermediate property that is proved of $\pi$, so the question remains open.
