Dimensions and the probability of finding odd numbers in Pascal's triangle and its relatives

It is well-known that the subsets of Pascal's triangle consisting of numbers not divisible by a prime p are relatives of the Sieroiski gasket. By computing the dimensions of these subsets, we obtain the puzzling result that in the infinite Pascal's of these subsets, we obtain the puzzling result that in the infinite Pascal's triangle, the probability that any number is not divisible by p is 0, for all primes p. By a more delicate analysis using hierarchical iterated function systems, we show the divisibility result is true when the prime p is replaced by any positive integer r. We give examples of similar results for Pascal's triangle generated by other polynomials.