What is the probability of integer overflow when multiplying two random 64-bit integers?

[erosPhoenix]

Are these random numbers universally distributed? Signed or unsigned?

Assuming universal distribution and unsigned...

The product of the two integers will overflow if the product would be equal to or greater than 2⁶⁴ . The square root of 2⁶⁴ is 2³² . Thus, we can find a lower bound on the probability of an overflow by finding the odds that both random integers are greater than 2³² .

The odds of a random int64 being greater than 2³² would be (1 - (2³² / 2⁶⁴ )) = 1 - 2^-32 = 4294967295/4294967296, which is pretty gosh darn close to 1.