Step 1: Define radioactive decay mathematically.
The number of undecayed nuclei at time $ t $ is given by: \[ N(t) = N_0 e^{-\lambda t} \] where $ N_0 $ is the initial number of nuclei and $ \lambda $ is the decay constant. Radioactive decay is a random, spontaneous process that begins immediately.
Step 2: Define half-life from the decay equation.
The half-life $ T_{1/2} $ is defined as the time when half the original nuclei remain: \[ \frac{N_0}{2} = N_0 e^{-\lambda T_{1/2}} \implies e^{-\lambda T_{1/2}} = \frac{1}{2} \] Taking natural logarithm: \[ T_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda} \]
Step 3: Evaluate option (1).
"Half the time needed for a sample to completely decay" is incorrect because radioactive decay is asymptotic - theoretically, $ N(t) \to 0 $ only as $ t \to \infty $, so there is no finite time for complete decay.
Step 4: Evaluate option (2).
"Half the time a sample can be kept before it starts to decay" is incorrect because radioactive decay begins spontaneously and immediately from the moment the sample exists. There is no waiting period before decay.
Step 5: Evaluate option (3) - the correct answer.
"The time needed for half a sample to decay" correctly restates the definition: $ T_{1/2} $ is the time for $ N_0/2 $ nuclei to decay (so that only $ N_0/2 $ remain undecayed). This matches our mathematical derivation.
Step 6: Final answer.
\[ \boxed{\text{Half-life = time needed for half a sample to decay}} \]