Probability question with intuitive answer - odds two travelers overlap at the same time?

Question

Let's say Person A sets out from Point X at 8:00 AM on Day 1 and travels for 12 hours until he reaches Point Y at 8:00 PM. Person B sets out from Point Y at 8:00 AM on Day 2 and travels for 12 hours until he reaches Point X at 8:00 PM.

If, for example, both Person A and Person B had constant rates of travel, we know that both A and B were in the exact same location (halfway between Points X and Y) at the same time of day - 2:00 PM (halfway between 8:00 AM and 8:00 PM).

However, let's assume we do not know anything about the pace of travel on either Day 1 or Day 2. Person A could have ran, walked, or rested such that at a given time between 8:00 AM and 8:00 PM, he could be anywhere between Points X and Y; the same thing applies to Person B. Additionally, A's pattern of travel on Day 1 does not have to be the same as B's pattern of travel on Day 2.

Question: What is the probability that at some specific time of day, A and B were located in the exact same spot (A on Day 1 vs. B on Day 2)?

This question is supposed to have an intuitive answer that is easy to understand without any complex modeling, but the answer is not clear to me since we don't have any information about the the rates of travel. Please help!

Answer 1

The probability is $0$.

It seems that we are modeling the time at which $A$ and $B$ are at the same location as a random variable. (This is unlike the first situation, where $A$ and $B$ travel at constant rates, which is deterministic.) Thus, this is a continuous random variable. And the probability of a continuous random variable having any specific value is $0$.

Answer 2

The question by Ian in the comments is crucial, as it changes the probability from 0 to 1.

As NicNic8 answered the question, it deals with being given a specific time beforehand, and then checking only at that time. That's I think how NicNic8 treated "specific".

OTOH, with the usual way the puzzle is given, there is no specific time given beforehand, it just asks if there is any time where both $A$ and $B$ are in the same spot, on their given days of travel.

The answer for that is that this happens surely (probability is equal to 1). The intuitive answer is that if both persons would have travelled on the same day, but with whatever schedule of running, walking or resting that they chose, they would have met (being in the same position) at some time in some place.

I'm assuimg that while paraphrasing the question, you reworded it and included the "specific" qualifier for time, which change the question significantly.

Answer 3

I believe there is a little missing information, do the two follow the same path? Based on your comment about traveling the same speed, I will assume the two travelers follow the same path. Also, it seems the fact that they are traveling on different days does not matter, and is there to convolute the problem. It just asks if the two are in the same position at the same time on their respective day. I’m answering from my phone, so sorry for the images being unpolished.

So assuming that they follow the same path, the probability is 1. Why?

For simplicity lets call position X = 0 and position Y = d, where d is the total distance of the path. We can also call the start of their journey (8 am) as time t = 0, so 12 hours later (t = 12) we know they’re journey is over. Looking at traveler A, he is at position 0 at time 0, and must be at position d at time 12. The opposite is true for traveler B, at t = 0 he is at position d and at time t = 12 he must be at position 0. Let’s look at a plot with your example.

In the graph below: The purple line is traveler A’s position at each point in time. The green line is traveler B’s position at each point in time. Both assume the travelers move at constant speed, and we see they intersect at position 0.5 at t = 6. This is with d = 1.

Now, see that an intersection means they are at the same position at the same time. From your example, we see an intersection. Now let’s assume they travel at any speed they want, maybe speeding up, slowing down or even backtracking. That means potentially one can draw any kind of green line and purple line. However, we still know the purple line starts at 0 and finishes at d, while the green line starts at d and finishes at 0. So, to see that the probability is 1, you can see that no matter how you draw these lines that they will always intersect at least once. Since the lines always intersect at least once, we know the probability of the travelers being in the same position at the same time is 1.