If every human being alive today (~8 billion) had a valid Visa card (14 digits not counting the beginning 4 and the check digit), there would be a 0.008% chance of a collision.
Also, not every valid number will be used (e.g. all 0s won’t be an option), and every number don’t need to be valid at the same time. If I renew 25 cards, their numbers are burned with no reuse.
That’s a long way to say, I’m not a fan in general of throwing in naive probability calculations and calling it a day.
The BIN/IIN is traditionally the first 6 digits. Extended BINs can be 8-11 digits, which is like subnetting -- the BIN sponsor can delegate assignment control of an extended BIN range to another entity. So in some cases, there can be as few as 4 "random" digits in the full card number (PAN).
E.g.:
BIN "random" Check
411111 111111111 1
ExtendedBIN "random" Check
41111111111 1111 1
We can't know the likelihood of hitting a valid number without knowing the count of assigned PANs in that BIN, but clearly the capacity would be 1 billion (or 10 million) possibilities.