Suppose that A and B take turns in tossing a biased coin which lands heads with probability p. Suppose that A tosses first.
a) What is the probability that A tosses the first head?
b) What is the probability that B tosses the first head? For both a) and b) above, find formulae in terms of p and sketch graphs.
No matter what the value of p, A is more likely to toss the first head than B. To try to compensate for this, let A toss once, then B twice, then A once, B twice, and so on.
c) Repeat a) and b) with this scheme. Give formulae and graphs.
d) For what value of p do A and B have the same chance of tossing the first head?
e) What approximately, is B's chance of winning for very small values of P? Give both an intuitive explanation and an evaluation of the limit as p → 0 by calculus.