A fair coin is tossed 100 times. The probability of getting tails an odd number of times, is

$\frac{1}{2}$

Let X denote the number of tails. Then , X is a binomial variate with parameters n = 100 and p = $\frac{1}{2}$ such that

$P(X=r)= {^{100}C}_r \left(\frac{1}{2}\right)^{100}, r= 0, 1, 2, ..., 100$.

∴ Required probability

$P(X=1)+P(X=3)+...+P(X=99)$

$=\left(\frac{1}{2}\right)^{100}\begin{Bmatrix}{^{100}C}_1+{^{100}C}_3+...+{^{100}C}_{99}\end{Bmatrix}=\frac{1}{2^{100}}×2^{99}=\frac{1}{2}$