Example: 3  
What is the minimum number of times that a fair coin must be tossed so that the chances of getting at least one Head are greater than ?

Solution: 3  
In a sequence of tosses , the probability of obtaining a Tail on every toss is
because at every toss, the probability of getting a Tail is , and also, all tosses are independent of each other.
Thus,
We want to be greater than , or . Thus,
Thus, a minimum of tosses are required.

Example: 4  
Two persons and are playing a game: They throw a coin alternately until one of them gets a Head and wins. How advantageous is it in such a game to make the first throw?

Solution: 4  
Suppose that makes the first throw. Let us calculate the probability of winning the game.
Let , denote a Head and a Tail respectively obtained by . A similar notation follows for . Now, will win the game in the following (mutually exclusive) sequences of tosses:
Thus, the probability of winning the game is
This means that one who makes the first throw has twice the chance of winning than the other
