Question

A coin is tossed n times. The chance that the head will present itself an odd number of times is $1/k$. Find the value of $k$.

Answer 1

In one throw of a coin, the number of possible ways is 2 since either head (H) or tail (T) may appear.
In two throws of a coin, the total no of ways is $2\times 2$ i.e.4 since corresponding to each way of the first coin there are 2 ways of second.
Similarly in three throws of a coin, the number of ways is $2^3$ and
thus in n throws, the number of ways $N=2^n$
And M = the favorable no.of ways =the number of ways in which head will occur once or thrice or 5 times or 7 times and so on,
${=}^n{C}_1{+}^n{C}_3{+}^n{C}_{50}+......=2^{n-1}.$
Hence required probability $p=\frac{M}{N}=\frac{2^{n-1}}{2^n}=\frac{1}{2}.$