Nuclear physics and half life of a radioactive element Half-life for certain radioactive element is 
5 min. Four nuclei of that element are observed 
a certain instant of time. After five minutes 
Statement-1: It can be definitely said that two 
nuclei will be left undecayed. 
Statement-2: After half-life i.e.5minutes, half of 
total nuclei will disintegrate. So only two nuclei 
will be left undecayed.
(A)Statement-1 is true, statement-2 is true and 
statement-2 is correct explanation for 
statement-1.
(B)Statement-1 is true, statement-2 is true and 
statement-2 is NOT the correct explanation for 
statement-1 
(C)Statement-1 is true, statement-2 is false. 
(D)Statement-1 is false, statement-2 is false
The correct answer for this question (D) .  What is the reason? 
I approached the problem in this way
After one half life , exactly half of the undecayed atoms will be left and this only depends on intial number of undecayed nuclei . So the correct answer according to me must be (A)
 A: "D" is the correct answer, as they are definitive statements and the physical situation is entirely probabilistic.
Let's assume "A" is True. In 5 minutes, 2 of 4 atoms must decay.
Which 2 atoms will decay? They are identical particles (in a quantum sense--which is far more strict then classically "being the same").
What is different about the 2 that will decay vs. the 2 the won't?
When will they decay? If the 1st decay does not occur instantly, let's say it takes 2 minutes and 1 second, another grad student comes in at the two minute mark and says, "A-Ha! 4 atoms. In 5 minutes their will be 2", but the 1st student says, no, there will be 2 atoms in 3 minutes.
Likewise, what if we split the atoms up. one student gets 2 and the other gets 2. Answer "A" means after 5 minutes, each student will have 1 atom. Do they both decay at exactly 5m? What if student 1 got the 2 the were going to decay? Then he says, "hey, both my atoms are gone after 5 minutes. The half-life is less than 5m" and the other says "I still have both my atoms, the half life is longer than 5m".
Who is right? Statement "A" is fraught with logical inconsistencies.
Now the idea that it only hold true for a large number of atoms is also incorrect. If you have 2 mol of atoms, in one half life the odds that you have exactly 1 mol left is tiny. Miniscule. (On the order of $1:\sqrt{N_A}\approx 1:10,000,000,000,000\ $).
The physics at work here is that all atoms are identical--meaning that they are not just the same, but the cannot even be distinguished from one another. By nature, and certainly not by grad students. Moreover, they have a constant probability per unit time of decaying. They key being: constant. No matter how new they are or how old they, it is the same.
If you understand the following it should make sense: A U238 atom that was part of Earth's formation (4B years old, that you dug up) and one that you made this morning in your reactor are IDENTICAL. The "old" one has no more likelihood of decaying than the "new" one.
A: Imagine the following experiment:
I have two buckets; in one bucket there are N balls. Every 5 minutes, I take each ball in turn; I toss a fair coin, and if it comes up "heads" I put the ball in the other bucket. If it comes up "tails", I discard the ball.
How many balls will there be in the second bucket after five minutes? On AVERAGE, there will be N/2 (as for each of the N balls, the probability of being discarded is exactly 50%). In reality, we know from the binomial distribution that there is a chance I have 0, 1, 2, 3 or even 4 balls.
In a radioactive sample, each nucleus can be thought of as one of these balls, and the passage of time is the "tossing of a coin". But instead of tossing a fair coin once per half life, we actually "toss a coin" with a VERY small probability of coming up tails, a great number of times - so that the cumulative probability after one half life is exactly 0.5. This results in the observed number of decays following the Poisson distribution. When the population becomes very large, it will look like "exactly half" decayed - but in reality if there are initially 2N atoms, then after a half life the number left will be $N±\sqrt{N}$. This means the relative error is $\frac{1}{\sqrt{N}}$, and as $N$ becomes very large, that error becomes vanishingly small. But when N=4, that's a big uncertainty...
