The statistical properties of the decimal digits of pi and numerous other irrational numbers, all of whose decimal digits go on without apparent pattern, have been studied. They appear to be truly random, in the sense that each digit appears approximately 1/10 of the time; each specific pair of adjacent digits appears 1/100 of the time, and so on. So without computing the 900,000,000,000,008th digit it’s reasonabe to declare that the digit being a 5 has a probability of 10%.
You could argue that it’s not actually random but pseudo-random, because ultimately the digits are completely determined by the underlying mathematics. Still, we can only observe behavior and measure distributions.
Thus any finite sequence of digits (let’s say a string of a million zeroes, or tomorrow’s winning lottery combo) not only can occur but must occur in the unending string of digits of any irrational number. Moreover it must occur an infinite number of times !
Computing all those digits is itself a non-trivial problem in applied math and computer science, and better algorithms are always being sought. There are many well-known infiinite series that converge to pi but they generally converge too slowly to be practical.
A breakthrough came a few years ago with the discovery of an algorithm to directly compute any specified digit of pi (like in your question)—except that it applies ony to hexadecimal digits, not decimal digits as you’d like.