Assume we take Pi out to an infinite amount of decimal places. We cannot determine the last digit of Pi, because there is no last digit, the string of random numbers goes to infinity. However, because there is no repeating pattern in the decimal portion of Pi we can assume that all numbers are equally likely to be the next number in the sequence as the length of the decimal portion goes to infinity. This in effect defines the next number in the infinite sequence as a random event. This means that each number is equally likely to be the next number so each has a 1/10 chance. Therefore, the occurence of each digit should be equal once we reach an infinite number of decimal places.
This is supported by looking at the differences between the occurence of each number over time as compared to the total number of occurences. The percentages shrink rather rapidly as the order of magnitude of the number of occurences increases.
The implications of this on the upcoming digits in the sequence are rather interesting, as this would mean that zero is actually more likely to occur than the other numbers, something of a paradox considering the random nature of the decimal place string.
-- (September 27, 2004) on Frequency of Each Digit of Pi