You're tossing a pair of coins: The probability of both ending up in the same state is 1/2--it's two out of four possible outcomes, HH, TT, HT, TH. But if the coins are identical, how do you distinguish between HT and TH? Well, no two coins are identical. Even the tiniest of imperfections will make them different in the eyes of Nature.
But what if you do this experiment with a pair of helium atoms? You can't have any imperfection in an atom--they are truly identical. The analog of coin-tossing is to give them the choice of two equally probable quantum states. It turns out that the probability of two helium atoms ending up in the same state is not 1/2, but 2/3. There are only three possibilities: both in the first state, both in the second state, and one in each state. This is truly bizarre! It's as if there was only one helium atom multitasking for two.
Because of this weird statistics, called the Bose-Einstein statistics, helium atoms (or bosons, in general) tend to aggregate, as if they were attracted to each other. This can sometimes lead to macroscopic effects, like super-liquidity in helium, coherent light in lasers (photon are bosons), or superconductivity (bosonic Cooper pairs).
Electrons are even weirder--they obey the Fermi-Dirac statistics. Two electrons (or fermions, in general) can never be in the same state. It's as if the result of tossing two coins was never two heads or two tails. They resist very strongly if you try to squeeze them into the same state. On a macro scale, this is why we have neutron stars. Neutrons are fermions, so even the tremendous gravitational pull can't make the neutron star collapse (unless it forms a black hole, in which case all bets are off).
@BartoszMilewski It's funny when you hear that oft-repeated science "fact", that the atoms in your body are replaced at such and such a timescale. Leaving isotopes aside, this seems like an empty statement or an impossible claim.
The important part in the original post is that we're giving the two helium atom a choice of two _same for both_ states. If one of them is here~ish and the other is 1km~ish away, they are nearly perfectly distinguishable (which does not mean that they aren't identical).
Coming back to "atoms in the body are replaced", this is most obviously effectful for carbon: C14 is unstable, so as long as that replacement is going on you have around the same ratio of C12 to C14 as the environment around you. Once you die, as long as you don't decompose (and thus stop exchanging carbon with the environment), the amount of C14 will go down as it decays, which is what makes carbon dating possible.
