Radioactive Decay - Fraction Remaining

Radioactive decay reduces the number of surviving nuclei by half during every half-life, so after $n$ half-lives the remaining fraction is ${\left(\frac{1}{2}\right)}^n$. This geometric halving follows from the exponential law $N=N_0{e}^{-\lambda t}$ evaluated at integer multiples of the half-life. For three half-lives, $n=3$, giving ${\left(\frac{1}{2}\right)}^3=\frac{1}{8}$ of the original sample. The value $1/6$ is wrong because it treats the decay as if three successive halvings simply add denominators ($2+2+2$) rather than multiply. The value $1/3$ incorrectly assumes one-third decays per half-life. The value $1/16$ corresponds to four half-lives, not three. Crucially, the decay is a statistical process governed by probability, so it is impossible to predict when any individual nucleus will disintegrate, only the average behaviour of the large population. This is the standard NCERT half-life treatment of exponential decay. A plausibility check confirms the sequence: after one half-life $1/2$ remains, after two $1/4$, and after three $1/8$, each step exactly halving the previous amount.