The rule from rung 2 is easy. What costs marks is naming the complement wrongly — and missing that the complement is often the easy thing to count.
Trap One: the Complement of "at Least One" Is "none"
This is the big one. The opposite of "at least one six" is not "no sixes and definitely some other rule" — it's simply "no sixes at all". "At least one" means one or more; the only way to fail that is to get exactly zero. So the complement of "at least one" is always "none". Likewise the complement of "none" is "at least one". Mix these up and your whole answer flips.
The Shortcut: Count the Easy Side
In the toy, counting "at least one 6" head-on is a pain — the winning cells are scattered all over the 6×6 grid (there are 11 of them). But its complement, "no 6", is a tidy 5×5 block in the corner: 25 cells, easy to count. So work out the easy one and subtract: P(at least one 6) = 1 − P(no 6) = 1 − 25/36 = 11/36. Same answer, a fraction of the counting. This is why the complement matters — it's not just a definition, it's a tool.
Trap Two: It's Still Between 0 and 1
Subtracting from 1 doesn't break the usual rules. The event and its complement are both real probabilities, so each sits between 0 and 1 (or 0% and 100%). If a complement comes out as 1.2 or −0.3, you've slipped somewhere — most likely you subtracted from the wrong number. Sanity-check: the two should add to exactly 1, never more, never less.