Probability of decay of a nucleus 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.
 A: 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).
A: What does "What is the probability that a nucleus disintegrates after 2 half lives?" mean?
The question is horribly worded.  I can come up with several interpretations, and there are probably more:
"What is the probability that a given nucleus does not decay until 2 half lives have passed?"  Since by definition we know half of the nuclei will decay in the first half-life, and half of the remaining half in the second half-life, the probability is 1/4 as @Ivella says.  
"What is the probability that a nucleus that didn't decay in the first two half-lives decays during the third?" the given answer of 1/2 is correct. 
"What is the probability that a given nucleus that didn't decay during the first half-life also does not decay until 2 half lives have passed?" the given answer of 1/2 is correct. 
"What is the probability that a nucleus that didn't decay in the first two half-lives decays after that time?"  100%, but it will literally take until the end of the universe for them all to actually decay.
A: What an annoying exam question! As Emilio says, your reasoning & calculations are correct. 
As your calculations indicate, if an isotope has a half-life of 4 days, then at any point in time a given nucleus of that isotope has a probability of .5 of decaying some time in the next 4 days, and a probability of .75 of decaying some time in the next 8 days, a probability of .875 of decaying some time in the next 12 days, etc. 
So how can we make sense of the reason quoted at the end of your question?

Disintegration of each nucleus is independent of any factor. Hence, each nuclei has the same chance of disintegration.

There's a tiny chance (1/1024) that a nucleus of that isotope will survive 40 days, but that nucleus still has a probability of 0.5 of decaying some time in the subsequent 4 days.
A: Read literally, my answer might be "infinitessimal" and requiring of a definition of the minimum quantity of time sufficient to observe whether a nucleus has disintegrated or not. (The Heisenberg uncertainty principle might feature). 
It is a quite appallingly badly worded question. The thought that it featured in an exam and that marks might have might have been awarded for the least obviously correct answer is sickening.
A: The above question is from an Indian Joint Entrance Examination. The answer has been challenged and the key has been corrected to 3/4. So OP's approach is right (atleast one of the possibility correct ones).
NOTE: I also came across is question in an IITJEE Archive and asked my doubt to a faculty. 
