The template uses a closed interval [lo, hi]. lo <= hi is the loop condition because the interval is inclusive on both ends, and lo = mid + 1 / hi = mid - 1 shrink it without ever revisiting mid. Computing mid = (lo + hi) // 2 is safe in Python (ints are unbounded), so you do not need the lo + (hi - lo) // 2 trick that C and Java need. Test against empty arrays, single-element arrays, the first and last elements, and out-of-range values: every binary-search bug shows up there.

The lower-bound template uses the half-open interval [lo, hi). The condition flips to lo < hi and hi = mid (no -1) because hi is exclusive. The function returns the first index where nums[i] >= target, equivalently the leftmost spot you could insert target and stay sorted. For absent target, this is exactly the result you want: where it would go. bisect.bisect_left from the standard library is the same thing, so reach for the import in production code and use the template only when you need a custom comparator.

The upper bound is the left edge of 'strictly greater' and equals the right edge of 'equal'. The only line that changes from the lower-bound template is < becoming <=. Subtracting bisect_left from bisect_right gives the exact count of items equal to a target in O(log n), which is the canonical 'count occurrences in a sorted array' answer and the classic interview follow-up to plain binary search. For absent targets, both bisects return the same index and count_equal correctly returns 0.