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Things seem to happen or they don’t. If a dice rolls on 6, does this mean that it could have been possible for it to land on 1-5? We seem to differentiate this kind of event from an “impossible” kind of event such as the dice spontaneously turning into a bird.

But how do we know that a dice that has already rolled a 6 could have rolled on any other number? If not, doesn’t that put both the event of a dice rolling on say 1 vs a dice turning into a bird into the category of impossible events?

How does this influence our probability calculations? We say that the probability of a dice landing on 6 is 1/6 since there were 6 possible outcomes. But if it couldn’t have landed otherwise, shouldn’t there be no other possible outcomes?

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    Possible events are models of events with imposed restrictions. Which restrictions are imposed depends on known laws of nature, knowledge of context, and/or assumptions one decides to make based on them, including simplifying assumptions. Not on what's "in the eye of God". Probabilities depend on the same. But you already know that, so why the endless stream of pseudo-questions belaboring the same over and over again?
    – Conifold
    Jun 9 at 1:31
  • Technically speaking it's not completely impossible for the die to spontaneously turn into a bird. With unimaginably low probability, all the particles in the die and surrounding materials could quantum tunnel into a new shape. Or, maybe the die could turn into a bird entirely by magic; we can't be absolutely 100% certain magic isn't real. Or maybe the laws of physics include an exception for this particular die. Or maybe the universe is a simulation and the programmer decided to play a prank.
    – causative
    Jun 9 at 2:05
  • @causative Saying that something is not completely impossible is a claim.
    – thinkingman
    Jun 9 at 3:13
  • @thinkingman Of... course it's a claim? Your point?
    – causative
    Jun 9 at 3:15
  • @causative What I mean is saying that something is not impossible requires knowledge that it is not impossible. But there's no evidence that a die turning into a bird is even possible. I suppose what you mean is that you can't prove that it is impossible?
    – thinkingman
    Jun 9 at 3:22

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What is possible depends on your perspective and how much you know. First consider what appears possible before the die has been rolled. If you know the exact physical configuration of the die (and the table, air, etc) as it begins to roll, then in theory you could calculate what number it must come up as. So from that perspective, only one number is possible, because you can cognitively rule out the other numbers.

But if you don't know the exact physical configuration of the die, then you have no way of predicting what it will come up as. From that perspective, any of the numbers are possible, because from your limited knowledge, you can't rule any of them out.

But what about after the die has finished rolling and you know it's a 6? At that later time, you can rule out the other numbers. However, when you talk about what "was possible," we can interpret this as what couldn't be ruled out based on a perspective P prior to the die being rolled. You can rule out the 1, 2, 3, 4, or 5 now, but you couldn't have ruled them out prior to the roll, so it is in that sense that we say those rolls "were possible."

Now, there are different perspectives P a person might have, prior to the die being rolled, and therefore there are different answers for what was possible. When you say what "was possible," you are implicitly invoking a certain P. I mentioned the perspective P1 in which a person knows all the physical parameters of the die and can calculate the physics of the roll, and the perspective P2 in which a person does not have that knowledge. Usually when we talk about rolling a die, it's understood P2 is the implicit perspective invoked. From P2 any roll is possible. But you could, if you specified, say that it was impossible for the die to come up anything but 6, based on a perspective of P1. These are not in contradiction, because they each only refer to a certain perspective. "'5 is impossible' follows from P1" does not contradict "'5 is possible' follows from P2."

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  • Interesting answer. How does one compare the probability of well defined event like a dice roll and another event that seems hard to probabilistically define then? (such as seeing a woman walking on the sidewalk). There seem to be two problems for the latter in my head: 1.) does probability make sense here? and 2.) depending on how you characterize the event ("woman walking on the sidewalk"/"52 year old woman who's 5'6 walking on the sidewalk"/"woman walking on the sidewalk at 7:30 PM"), there seem to be multiple answers here.
    – thinkingman
    Jun 9 at 3:21
  • +1 A fair coin is a good example. It is always assumed to be a system completely described by two states. But a nickel is wide enough to include a third state (on edge) and there is a non-zero probability of occurring. If you only assume two states and the third state occurs then it could be mistakenly observed as a "miracle"
    – user64314
    Jun 9 at 15:57

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