Question

In a game “odd man out” each of $m\ge 2$ person tosses a coin to determine who will buy refreshments for the entire group. The odd man out is the one with a different outcome from the rest. The probability that there is a loser in any game is.

• A. $\frac{1}{2^{m-1}}$
• B. $\frac{m-1}{2^{m-1}}$
• C. $\frac{m}{2^{m-1}}$
• D. none of the above

Answer 1

The correct option is C

$\frac{m}{2^{m-1}}$

Explanation for the correct answer:

Find the probability of a loser in any game :

Let $S=$ the sample space, then $n\left(S\right)=2n$

if a person is an odd man out if he alone gets head or tail.

So the number of ways in which the person is precisely getting one tail (or head) and the remaining are heads (or tails) $={}^m{C}_1=m.$

So, $m+m=2m$

Therefore the total number of favorable cases $n\left(E\right)=2m$

$\therefore$The probability of a loser in any game is,

$\begin{array}{rcl}P\left(E\right)& =& \frac{n\left(E\right)}{n\left(S\right)}\\ & =& \frac{2m}{2^m}\\ & =& \frac{m}{2^{m-1}}\end{array}$

Hence, the correct option is C.