Question:
A given sample of radium-226 has a half life of 4 days. What is the probability that a nucleus disintegrates after 2 half lives?
Attempt at solution:
After time $t$, the number of nuclei remaining will be $N = N_0e^{-kt}$ where $k$ is the decay constant, $k= \ln(2)/T_{1/2}$ , here $T_{1/2}$ is the half life, and $N_0$ is the initial number of nuclei. This implies $N$ nuclei have survived so far, hence , probability of survival $P= N/N_0$ , and hence probability of decay is $1-P$. Putting $t=2T_{1/2}$ in the equation, we get the required probability as $3/4$.
However, the given answer is $1/2$, the explanation provided was,
Disintegration of each nucleus is independent of any factor. Hence, each nuclei has same chance of disintegration.
I cannot understand this statement and also I don't understand what is wrong in my approach.
Any help is appreciated.
Best Answer
Your calculations and reasoning are correct, and the stated answer in your quote is wrong. Find a better textbook (if this is a practice problem) or report this error to your instructor (if it came from class materials).