If you were to read tomorrow's newspaper it is intuitively more likely that the whole number 1 would appear more times than 643689443. Is there an expected distribution of numbers used in general?
I'm not sure how to express this in mathematical terms.
On
Not sure if this answers your question, but you might be interested in Benfords Law. Basically, smaller digits are more likely to appear than larger digits. $$P_1 \gt P_2 \gt P_3 \gt P_4 \gt P_5 \gt P_6 \gt P_7 \gt P_8 \gt P_9$$
Your intuition is correct. Take a look at Benford's Law.
It only applies to the first digit in a number. But, as mentioned in the article, there are generalizations to longer strings.