Let’s start with a quick riddle:
(Riddle A) I flip two coins. If the first gives heads, what is the chance that one of the two coins gives tails?
This one’s pretty simple. The answer is 1 in 2, of course.
Another riddle, very similar to the first:
(Riddle B) I flip two coins. If one of them gives heads, what is the chance that one of the two coins gives tails?
Surprisingly, the answer to this riddle is 2 in 3. Let’s try to understand why the answers are different to the two riddles. The key seems to be that in Riddle A I specified which coin I was talking about, while in the Riddle B I did not. To formally analyze this problem, let’s think of all the possibilities after I flip two coins. We can have HH, HT, TH, or TT, where each letter specifies which side each of the two coins landed on. The ordering is important: the first letter corresponds to the first coin, and the second letter to the second coin.
In the first riddle, we are told that the first coin gives heads. Then, the only two remaining possibilities are HH and HT. Thus, the chance that we get a tails is one in two.
In the second riddle, we are told that one of the two coins gives heads. So, the allowable possibilities are HH, HT, and TH. In this case, the probability of getting a tails is two in three (there are two cases where we end up with a tails).
As a quick side-note, don’t be worried if you weren’t able to figure this out. I wasn’t able to either, until I learned the correct answer. This problem is similar to the historic “Boy or Girl paradox”, first presented in Martin Gardner’s Mathematical Games column in Scientific American in 1959 and popularized by Marilyn vos Savant (who’s officially the person with the world’s highest IQ, if that makes you feel any better). If you weren’t completely convinced by my explanation, you can find plenty of people online talking about this problem, and I encourage you post the riddle somewhere online and enjoy the battles that you start with it. The rest of this article is for people who agree with my explanation or are willing to suspend their disbelief.
Now let’s consider the deeper implications of this problem. It should be clear that the difference between the two riddles has to do with distinguishing between the two coins. Although both riddles ask about properties that don’t depend on which coin was which at the end – they ask for the chance that either of the two coins shows tails – Riddle A provides information that does distinguish the two coins – the first shows heads – while Riddle B does not ever distinguish between the two coins – one of them shows heads. In the second riddle, we can say that the two coins are identical.
Two objects are identical when we can’t perform an experiment to tell the two apart. Normally, coins would never be identical. I can always scratch one with a knife or mark one with a Sharpie and easily tell them apart. I don’t even need to do anything to them; I can just map out a corner of one coin with a microscope and tell them apart because just by chance the two coins will wear down in different ways. But we can’t perform any of these experiments in a riddle, so we have to take the two coins to be identical in the true, physics-based sense until we get information that tells them apart. For example, learning that one of the two coins is “first” provides a label that we can tell them apart with. That’s why we get different answers to the two riddles: the coins in the second riddle are identical while the coins in the first riddle are distinguishable.
The idea of identical objects has deeper implications in physics. One of the fundamental postulates of quantum mechanics, our current best theory of how small things in the universe work, is that fundamental particles (like electrons or photons) are identical. Every electron is the same, no matter where it is or what’s around it. We cannot tell apart the “first” electron from the “second” electron because we cannot label electrons by cutting marks in them or sticking anything to them.
In fact, quantum mechanics even predicts that the number of electrons does not need to be fixed: an electron can spontaneously appear and interact with other particles as long as it is safely disposed of in the end, almost as if the universe decided to take out a loan. If electrons are appearing and disappearing at ease, it’s pretty much inconceivable that you could keep track of a single electron for too long.
As another brief side note, if some New Age guru brings in quantum mechanics and particles appearing and disappearing to defend “I am one with the Universe”, try asking them the two riddles. If they get it wrong, they definitely don’t know what they’re talking about. If they get it right, they’re a physicist or mathematician who took a strange turn in life.
Because they are identical, fundamental particles like electrons can only exist in states that don’t care about which electron is which. For them, the states HT and TH (if they were like coins) don’t exist separately. Rather, the state looks like HT+TH, so that if you switch the first and second electrons (so that HT becomes TH and TH becomes HT), you get back the exact same state. States of the electrons with these properties are called symmetric under exchange because you can exchange the two electrons and get back the exact same state.
These combinations of states turn out to underlie why atoms tend to bond with each other. When two atoms come close enough together, we can’t tell apart their electrons. As a result, the electrons from both atoms form combined states (similar to the HT+TH state), centered between the two atoms, that hold together the molecule in a strong chemical bond.
If the idea of identical objects still seems a bit strange, there’s good reason for that. In the macroscopic world that we inhabit, distinguishability rules, and because humans evolved to fit best with their environment, indistinguishable/identical objects will always remain nonintuitive. But even if it isn’t intuitive, we can still study it mathematically (like the riddle I presented at the start), and through this formal system, we can still make predictions about our world in the realm where human intuition fails.