Probability of a false positive for a given image = p Probability of N false positives (assuming independence) = p^N Threshold N is chosen by Apple such that

Probability of a false positive for a given image = p Probability of N false positives (assuming independence) = p^N Threshold N is chosen by Apple such that p^N < 10^-12, or N log p < -12 log 10, or N > -12 log(10)/log(p) [since log(p) < 0, since p < 1]. ETA: Suppose, just for the sake of the argument, that p = 10^-3 (one false positive in 1000, so really quite bad). Then log(p) = -3 log(10), so N > -12 log(10)/(-3 log(10)) = 12/3 = 4. Similarly, if p is one in a million (10^6), then N would be required to be > 12/6 = 2. In practice, I'd expect N to be larger than 4, in other words, Apple being very conservative here. ETA: The above doesn't take into account how many images M you have. The analysis gets more complicated, but N needs to be way larger than 4. I'll think about it some more.