You and I play a game involving successive throws of a fair coin.
Suppose I pick HH and you pick TH. The coin is thrown repeatedly
until we see either two heads in a row (I win) or a tail followed
by a head (you win). What is the probability that you win?
You have two bags, four red balls and four white balls. You must
put all the balls in the bags although you are allowed to have one
bag empty. How should you distribute the balls between the two bags
so as to make the probability of choosing a red ball as small as
possible and what will the probability be in that case?
In how many different ways can I colour the five edges of a
pentagon red, blue and green so that no two adjacent edges are the
The concept of gambler's ruin is useful to include, where the winning system cannot be continued because a series of losses has caused the situation where there is nothing left to bet with.