Two coins, two dice, lengths and areas

With two coins or two dice, write out every outcome first, in order.

Two coins (or one coin tossed twice): HH, HT, TH, TT. That's 4 outcomes, and HT and TH are different.

A common mistake: 'there are 3 results (two heads, two tails, one of each), so each is 1/3'. Wrong! 'One of each' happens in two ways (HT, TH), so it has probability 2/4 = ½.

Three coins: list them in a fixed pattern, HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. That's 8 = 2³. With n coins there are 2ⁿ outcomes.

Two dice: draw a 6 × 6 grid, with the first die along the side and the second along the top. That gives 36 outcomes. (1, 4) and (4, 1) are different squares.

Count from the grid: a sum of 8 fills 5 squares, (2, 6) (3, 5) (4, 4) (5, 3) (6, 2), so P = 5/36.

Another common mistake: 'the sums 2 to 12 are 11 results, so each is 1/11'. The sums are not equally likely. A sum of 7 fills 6 squares, but a sum of 2 fills only 1.

'At least' and 'at most': 'at least one head' means 1 or more; 'at most one head' means 0 or 1. With two coins, at least one head is HH, HT, TH: 3/4.

When the outcomes can't be counted (where a spinning arrow stops, where a helicopter falls in a lake), use sizes instead: P = favourable length ÷ total length, or favourable area ÷ total area.