Q: What is the probability of an outcome after it’s already happened?

Physicist: There are a lot of subtleties to this.  Reading the question, your gut reaction should be “Duh, it’s 100%!  Wait, is this really a question?”.

And yet, there are many times in which you may find yourself estimating probabilities on things that have already happened.  If you flip a coin and cover it or go looking for a lost dog, the “true” probability is always 100%: the coin is definitely either heads or tails, and Fluffins (the wonder dog) has a 100% chance of being exactly where it is.

In liar’s dice all the bets and estimations are based on events that have already occurred, but are unknown.

Probabilities are usually defined in terms of the uncertainty in what’s known.  Liar’s dice is a beautiful example (so are most card games for that matter); all of the dice are what they are, and yet in the picture above, if you’re the player on the left, then there’s a chance of 1 that all of your dice are 5’s, but there’s an even chance that your opponent’s dice could be any combination.  From the left player’s perspective, there’s some chance that the dice on the right are, say, “1,2,3,4,5”, even though from the right player’s perspective, that chance is zero (the right player knows their dice are not “1,2,3,4,5”).

Long story short: probability is extremely subjective.  Whether an event happened in the past or will happen in the future doesn’t make too much difference, it’s the knowledge you have about an event that defines its probability (for you).  That said, in terms of gaining knowledge about an event, it helps a lot for it to have been in the past.  If you were some kind of time traveler it would be a lot easier to determine the result of a coin toss by just looking at it after it’s happened, rather than going to the trouble of predicting it before it’s happened.  That’s why there are Futures Markets, but not Pasts Markets.

But the spirit of this question is really about some kind of “objective probability”.  Maybe you don’t know how something in the past turned out, but surely if you somehow had access to all of the information in the universe you’d be able to determine that the probability is 100% or 0%.  Surely everything in the past either happened or didn’t, it’s just a matter of finding it out.

Very, very weirdly; no.  You have to root around in quantum mechanics to see why, but it turns out that even things in the past, in the most objective possible sense, are also uncertain.  This doesn’t mean that, for example, the Nazi’s may have won the war (since it’s pretty well-known that they didn’t), but it does mean that if an event is so small and fleeting that it leaves no real trace, then it may have happened in multiple ways (quantum mechanically speaking).

Answer gravy: This is high on the list of the weirdest damn things ever.

Way back in the day, the double slit experiment demonstrated that a particle (and later much larger things) can literally be in two places at once.  This means that the question “where did I leave my quantum keys?” doesn’t have a definite answer.  The probability that the particle will be found going through one slit or the other is non-zero, not just because the position isn’t known, but because it can’t be known (essentially, there’s nothing definite to know).  The first reaction that any half-way reasonable person should have is “dude, you missed something, and that particle totally has a definite position, you just don’t have a way to figure out what it is”.  But physicists, being clever and charming, found a way to prove that that isn’t the case.  It can be shown that, regardless of what you do or how you measure, quantumy things don’t have a definite position.  This is basically what Bell’s theorem is all about.

Not comfortable with reality being merely a little weird and uncomfortable, a dude named Franson proposed an experiment to demonstrate that the past is in a similar superposition of states.  Not only can things be in multiple places now, but they can do it at multiple times.

In the Franson experiment a photon is emitted at a random time and shot toward a beam splitter, which allows it to take one of two paths; a long path and a short path.  You’d “expect” that the photon would be emitted at a particular time, then take one (or both) paths, and then randomly exit (diagram below).  The thinking is that, since the two versions of the photon arrive at the second beam splitter at different times, there’s no way for them to interfere.

The Franson experiment: You’d think that a photon would be emitted at a definite time (left) and then move through the paths and exit randomly, either sooner or later.  But (right) we find evidence of interference at the exit, so the photon must exit at one time and enter at two.

However!  When this experiment is done (with a random photon source) interference is seen.  Therefore the photon must be arriving at the second beam splitter from both paths (similar to how the double slit experiment creates interference).  But that means that the photon must have been released at two different times.

There’s some subtlety that I’m not including, such as the fact that paths above are only half of the device (the other half is identical), and that the experiment requires entangled photons, but if you’re interested in the details you can read the original paper.

What’s really horrifying is that this experiment is done pretty regularly!  Nothing special.  The past genuinely is in multiple-states, and as a result the probabilities of events in the past can be damn near anything.

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10 Responses to Q: What is the probability of an outcome after it’s already happened?

  1. Anthony Rose says:

    Very interesting, especially that quantum stuff! Thank you. I think there is also another way of looking at it, not contradictory at all, just a different angle, and that is, the probability depends on the precise question you are asking, which can unwittingly be changed after something has already happened. For example, if 1,000,000 people enter a lottery, the odds of any randomly selected particular one of them winning (all other things being equal) is a million to one. However, once the lottery is over and person A has won, some people say that the million-to-one chance has indeed occured, at least for person A. However, the original question itself has now subtly been changed by the knowledge of the past event, from “what are the odds of any randomly selected particular person winning” to, “what are the odds of person A (the actual winner) winning”, which of course, is 1. Or, put less particularly, the question has become “what are the odds of any undefined one (non-selected before the event, selectable after the fact) of the entrants winning”, which is 1,000,000 out of 1,000,000, or 1 again. One can see the odds of the original question still hold unchanged even after the event, by randomly selecting any one of the million entrants to be the particular target of the question. It’s still a million to one that your random selection turns out to be the winner.

  2. Anthony Rose says:

    Hope I’m not being too chatty, but that last experiment really is fascinating. Why would the photon exit at one time and enter at two? What would make it ‘choose’ to do it that way round, than exit at two times and enter at one? That’s just too much of a coincidence… Surely it is more likely that the photon enters at a spread out multiple of times, and exits likewise, giving room for interference during the overlap? How long can the 2nd path be, and it still happen? If you could put the 2nd path’s mirrors on some distant stars, would there still be interference?
    Most of all I find it so fascinating because it is almost as though a ‘photon’/light-wave is not really something distinct/unique at all, but merely the effect produced by something else behind the scenes. Like making a wave with your hand under a blanket to knock something off the bed: the movement behaves like a particle and like a wave (well, sort of 🙂 – good enough for this poor analogy!) but is actually neither, because the bump and the wave are merely the effects or influence of something beyond either. As effects they can be staggered/spread across time and space more easily without losing their relationship to the original cause.

  3. The Physicist The Physicist says:

    In regular probability (this should have been said explicitly in the post) probabilities are changed by “conditionals” or “givens”. For example, “what is the probability that I’ll be at work given that today is Monday?”
    In quantum mechanics the same ideas show up. In the double slit experiment you know where the photon hits the wall, but you don’t know which slit it went through. All of the probabilities are calculated given that the photon hits the wall, having made it past the slits.
    In the Franson experiment the given is that the photon arrives at a particular time. Given that, the emission time can be either (both) of the early, possible moments.

  4. Will says:

    To put it another way; when an event happens in relation to an individuals past or future has nothing to do with the probability of the event. If the coin is heads that doesn’t mean that the coin had a 100% chance to land heads when you flipped it. It’s still a 50% chance, it’s just a 50% chance that has already occured.

  5. GenePoz says:

    I think a more interesting and concrete example of conditional probability would be if you explained the “Let’s Make a Deal” car vs. goats conundrum, where, given a goat behind 2 doors and a car behind 1, once the contestant chooses a door, the host opens the door where one of the goats is (of the 2 doors contestant has not chosen) and gives the contestant an opportunity to switch his choice. Intuitively we would think that switching gives only a 50% chance of getting the car, but it’s not. It’s 2/3 chance of getting the car.

  6. The Physicist The Physicist says:

    Done and done!

  7. Dan says:

    Is this kind of like that variation of the double slit experiment ? I saw an incredible experiment where they placed a measuring device behind the slits, but in front of the detector screen, and they didnt turn it on. When they fired single electrons through, they got interference as was expected. But this extra device was used in a really cool way, they would fire an electron, let it’s wave function pass through (both/neither/whatever) slits and carry on toward the screen as a wave with an interference pattern in it, however they timed it so that this second detector would activate after the wave had passed through the slits and detect it. Amazingly even though it had already gone through the slits and become a wave with interference, this late observation appeared to make the electron wave, backtrack along its own timeline, into the past, where it could then go through one or the other in order to maintain causality, and the electron then hit the screen without interference, in two lines. This post reminded me of that experiment, because, (and correct me if im wrong) if an electron has already interfered with itself by going through a double slit setup, then surely the probability of getting an interference pattern on the screen is 100%, but its not, because from what this experiment says, its still not certain, because if you observe that electron before it gets to the screen, it will turn back into a localised particle again.

  8. Marius de Jess says:

    The author tells us about the quantumy world.

    My question is how does this world affect us who are living in our world which is not the quantumy world.

    In our world a thing is in a definite location at a definite time or period of time, and if we don’t respect it but violate or disregard it we can lose our existence, i.e. kill ourselves or get killed by the inexorable laws of our world’s physics and our world’s laws of nature whatever.

    What does it matter to us that in the quantumy world a particle whatever can be in two or more places and times at the same what? time, and where? same place, or not all in time neither in space, i.e., does not exist — whatever?

    How does this piece of information or datum in the quantumy world affect us as to kill us, if we do not know about it at all and unwittingly if possible at all violate it; or how do we go against this quantumy situation in the quantumy world, where a particle can be in several locations in space and what, at the same time or even at different times, or it does not at all exist?

    Marius de Jess

  9. Frank Weyl says:

    Hi Dan,
    A photon doesn’t experience time. The very old and misguided quote that: ,The simplest answer is usually the truth’ does not and can not be associated with QM.
    Analogies have there rightful place. Pushing analogies to the limit does not give any fresh information either intellectually or physically. Analogous forms can be so radical that they cease to have any connection to the observed experiment or observation. Ad Hoc imaginings are okay if they remain at least within the parameters of the conjecture. Most times they are so diluted by explanation that the similarity of the observation etc, gets lost….
    The complex and entangled ‘hidden variables’ that we find in all our equations in chaos theory, at al, will be reveled when we have evolved ourselves sufficiently to either think it, or construct systems that will enable us to understand. We are only at the beginning of our intellectual grasp of reality, our neurology is evolving substantially. Our consciousness is only a shadow on a wall compared to the real and holistic ability of our evolving brains to truly understand our existence from a quantum event that was, without a creator, and truly majestic…

  10. Frank Weyl says:

    Hi GenePoz
    conditional probability? Isn’t that an oxymoron?
    Just asking. would like to know your conjecture on the above statement?

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