Tune r and b, watch the S-curve move.
MinHash gives each document a signature of length k. LSH splits that signature into b bands of r rows each, so k = b · r. Two documents are candidate pairs if they agree on at least one full band.
If the true Jaccard similarity is s, two signature entries agree with probability s. So:
P(agree on a band) = sr
P(disagree on every band) = (1 − sr)b
P(candidate pair) = 1 − (1 − sr)b
This last expression is an S-curve in s. Its steepest point — the approximate threshold — sits near t ≈ (1 / b)1/r Below that threshold most pairs get filtered out (good for dissimilar pairs); above it most pairs make it through (good for similar pairs). The playground marks the exact 0.5-crossing, which is numerically very close to the textbook approximation above.
AND/OR (standard banding) requires all r rows inside a band to agree, then any band can trigger a candidate. It amplifies the gap between low and high similarity — high similarity pairs almost always collide, low similarity pairs almost never do.
OR/AND is the dual: any row inside a band suffices (OR), but every band must fire (AND). The curve is a 180° rotation of AND/OR around the point (0.5, 0.5) — the two constructions are mirror images. So for a given (b, r) their thresholds sit close together, but their shapes differ: AND/OR pushes the low-similarity tail exponentially toward 0, OR/AND pushes the high-similarity tail exponentially toward 1. For near-duplicate detection you almost always want AND/OR — killing false positives hard matters more than extracting the last few nines of recall. The "overlay both" mode lets you compare them on the same (b, r).
Classical LSH theory (Mining of Massive Datasets, ch. 3) goes further: you can chain AND-of-OR-of-AND-of-OR… constructions to sharpen the S-curve arbitrarily, at the cost of more hashing per document.
Move r and b and watch two things at once: where the S-curve crosses 0.5 (the threshold), and how steep it is (how clean the cut is). Your choice of r and b trades off recall (catching true near-duplicates) against the number of candidate pairs you'll pay to verify.