How can probabilities be modeled in a universe where time travel is possible?

Question

Please don't take this as a joke, its actually a serious question. If it sounds silly its only because of my (lack of) understanding of probabilities, but my motivation is genuine.

Lets take the following 2 events:

• $A$: Shooting once at a bullseye and hitting at an specific point $P_A$.
• $B$: Shooting twice at a bullseye and hitting both times in the same point $P_B$.

Under classic asumptions (for some sensible definition of classic) both events have $0$ probability. Right?

Now suppose we have backwards time travel (despite physics contradictions). I can watch where the projectile hits the first time, then go back in time, and make a prediction with probability $1$ for $P_A$. However, even with time travel, event $B$ still has $0$ probability.

My questions are:

• Is there some real difference in how these two events are modeled, or interpreted, or is there (more probably) some logical flaw in my reasoning?
• Does this means that time travel is impossible from a logical point of view, without even looking at physics?

Answer 1

This is not a silly question at all. What you are wondering about is the dependence of the probability of an event on the protocol, and about the precise meaning of such claims as "the probability of even $E$ is $p$". A lot has been written about these issues.

Sometimes, probability is meant to reflect a frequency of occurrence. For instance, a typical interpretation of "the probability of rolling a 6 on an honest die is $1/6$" is that if you roll the die a lot of times, then about $1/6$ of the times you will have rolled a 6, and, moreover, the more times you roll, the closer the ratio will actually get to $1/6$ (kinda). This certainly agrees with our intuition acquired after playing with dice.

Other situation are quite different. For instance, "the probability that intelligent life exists outside of our solar system is 0.76" is not quite frequentists, but instead it is reflecting a certain state of knowledge. After all, either there is life out there or there isn't, so clearly the probability is either $0$ or $1$, we just don't know which.

Situation where the probability of an event seems to change drastically when the protocol changes do not require time-travel. A famous example is the Monty Hall Problem.

Mathematically, a common way to model probabilities is through measure theory, an approach initiated by Kolmogorov. However, there are other possibilities. For a very good discussion on the essence and foundations of probability I would recommend Jayness' Probability Theory - The Language of Science (I believe the first few chapters are freely available online.

Answer 2

You don't need time travel to come up with these kinds of paradoxa. For example, take a normal 6-sided die. Not the mathematically idealized variety, but a real, physically existing die. Prefererrably one with slightly rounded edges, as most dies have. Now throw the die. Theoretically, if you'd know the exactly speed and angular momentum of the die at the moment it leaves your hand, plus the pressure distribution of the air the die travels through, plus a fine-grained 3-D map of the underground it eventually hits, you can simulate the throw, and find the face that will point upwards before the die comes to a rest. Say that face is 5. Does that contradict that statement that $P(\text{Die shows 6}) = \frac{1}{6}$? No, it doesn't. It merely says that $$ P(\text{Die shows 6} | \text{Precise physical cirumstances of the row}) = 0 \text{,} \\ P(\text{Die shows 5} | \text{Precise physical cirumstances of the row}) = 1 \text{.} $$

The same goes for time-travel. Time-travelling just saves us from having to do all the tedious computations, because we can just let the real world work out the answer, and feed it back in time to us. And, rather unsurprisingly, we find that $$ P(\text{Die shows 6} | \text{Die showed 6}) = 1 $$ but that doesn't contradict $$ P(\text{Die shows 6}) = \frac{1}{6} \text{,} $$ because the two conditions are different.

Answer 3

I think it has to do with the probability of an event happening versus knowledge of that event.

Take scenario $A$. The probability of shooting any particular point is $0$. However the probability of hitting any point is $1$ (assuming you don't miss entirely). You just don't know which one. Time travel allows you to know which one, allowing you to make a perfect prediction. The crux of the event is in choosing the proper point.

In scenario $B$, the event itself has probability $0$. The first shot goes anywhere. Time travel lets you know where that will happen. The second shot goes anywhere, but the probability of it going where the first shot is is $0$. The crux of the event is in hitting the same point twice. Knowledge won't change those odds.

So to answer your first question. Scenario $A$ is about an event that will physically happen. You just don't know how it will happen without time travel. Scenario $B$ is about an event that is impossible, and time travel knowledge will not change that. If you were to rephrase the scenarios and add "given that you know exactly were they will go" for the time-travel version, it shows that the scenarios are different. Time-travel $B$ and non-time-travel $B$ having the same answer isn't a flaw, it's how it works in this scenario. Knowledge can't make an impossible event possible.

For the second question, I don't believe this particular example is a logical inconsistency any more than rewinding a tape and predicting what will happen on the second viewing would be.

Answer 4

A solution to the dilema is the butterfly effect.

If the probability of hiting point A in your future is near zero, and you wait until after the arrow hits A, and then travel back in time to before the shot was made and shoot again you will find (under this theory of time travel) the probability of hitting A is once again zero. That is, even allowing the possibility of time travel, the universe never unfolds the same way twice.

IMHO, this is the mechanism by which time actually works, any movement away from the present moment, whether forwards or backwards in time, makes it impossibly unlikely to return along the same path to the same circumstance. Thus from the time traveller's perspective all movement in time feels like forward movement.

Answer 5

First of all, if the bullseye and bullet have nonzero diameters, the probability of hitting the bullseye is not zero. Second, many people seem to have a penchant for using words to mean something different from their original use and claim the misuse to be an abstraction instead of what it is. This is the case with the word "probability." Now we have "empirical probability" which is an estimate of what or might not be an unknown probability, and "subjective probability" which is imagined, and "classical" or "theoretical probability" which is what probability originally meant and should still mean. This is a common problem in mathematics; e.g., people claim "mulltiplication" is abstract because they apply the same word to operations that are different from repeated addition. It is convenient to do that and not illogical but calling a space a diamond does not make it abstract. It merely makes it confusing.