For the problem of the points, let Pn := Pr An,n and r := p (1 − p). Use

(3.16) and (1.4) to show that

This formula is intuitively appealing in that it directly shows that Pn = 1/2 for all n if p = 1/2 while it increases (decreases) with growing n if p is larger (smaller) than 1/2. An alternative representation of (3.28) is easily seen to be

where q = 1 − p, which is also an attractive form.
(Contributed by Markus Haas)

P = p + P=P+P (* - 1)()..