Probability of 8 or 9 digit sequence colliding in the same place in two 65 digit numbers

Question

I have two numbers:

3032643431333337636238613038343231383364303731376566303037663231

3861663464383131656131653461343961343364303737663565356561653361

36430373 repeats from digit 36-45 (I only extracted the 8, but the 9th matched too).

What are the chances that the same 8 digits would repeat in two 65 digit length numbers? What are the chances that the same 9 digits would repeat in two 65 digit length numbers?

Minor Details: I extracted 8 digits from a hash(md5) that was converted to a binary string and then hex encoded to get the 65 digits. I found the collision and was curious as to the odds.

Thanks

Edit: As was commented, the digits must be between the 36th and 45th position. Not elsewhere in the number.

Answer 1

If you want them to be in a specific set of places, the chance of a match starting at a given place is $10^{-8}$ for $8$ digits and $10^{-9}$ for $9$ digits. Assuming independence and ignoring the chance of two matches (both excellent approximations), there are $58$ places an $8$ digit run can start, so the chance of an $8$ digit match somewhere (but in the same places in both numbers) is $58 \cdot 10^{-8}$ If you want any pair of $8$ continuous digits to match (not necessarily the same places in both numbers) you have $58^2$ pairs of runs, so approximately $58^2 \cdot 10^{-8}=3.364\cdot 10^{-5}$ chance of a match somewhere.

Answer 2

If "repeats" means also "on the same places", as it is suggested by your example, it is simply a chance of taking a fixed 8-digit number, because the other digits are arbitrary. Hence the probability is $10^{-8}$.

You are lucky or your numbers are not uniformly distributed. :-)