Question

A coin is tossed $n$ times, what is the chance that the head will present itself an odd number of times?

Answer 1

If coin is tossed $n$ times then possible number of outcomes $=2^n$

Number of ways in which head comes odd number of times $={}^n{C}_1{+}^n{C}_3{+}^n{C}_5.......$

Sum of odd coefficient in the expansion of ${(1+x)}^n$

$=\frac{2^n}{2}=2^{n-1}$

$\therefore$ desired probability $=\frac{2^{n-1}}{2^n}=\frac{1}{2}$