In a game called 'odd man out' , m (m > 2) persons toss a coin to determine who will buy refreshments for the

entire group. A person who gets an outcome different from that of the rest of the members of the group is
called the odd man out. The probability that there is a loser in any game is

Let A denote the event that there is an odd man out in a game. 

The total number of possible case s is $2^m$ .
A person is odd man out if he is alone in getting a head or a tail. 

The number of ways in which there is exactly one tail
(head) and the rest are heads (tails) is${ }^m{C}_1=m$
Thus, the number of favorable ways is m + m - 2m. 

Therefore, $P\left(A\right)=\frac{2m}{2^m}=\frac{m}{2^{m-1}}$