Equally likely outcomes

Outcomes are equally likely when every simple event has the same chance. If S has n outcomes each of probability p, the axioms force np = 1, so p = 1/n.

If n(S) = n and the event E has n(E) = m outcomes, then P(E) = m/n: the number of outcomes favourable to E divided by the total number of possible outcomes.

The formula needs equally likely outcomes. On the weighted die above, P(6) = 1/2, not 1/6.

Pick the sample space so its outcomes really are equally likely. For two dice, use the 36 ordered pairs (x, y), not the 11 sums 2 to 12: a sum of 7 arises from 6 pairs, a sum of 2 from only 1.

Mela token bag (made up): 5 red, 3 green and 2 yellow tokens of the same size. Each of the 10 tokens is equally likely, so P(red) = 5/10 = 1/2 and P(green) = 3/10.

Mela coin stall (made up): with three fair coins each of the 8 outcomes has probability 1/8, so P(exactly two heads) = 3/8 and P(at least one head) = 7/8.