# Thread: Recreational Mathematics

1. I'm not sure but I suspect it will greatly depend on what you assume about the distribution of possible run lengths. Also it probably depends on how long you observe them for. Obviously if you observer for the whole of your ourward journey and haven't turned when you do turn then they are definitely running further than you.

To make calculations easier lets just talk about the outward journey and say the run finishes when you turn. Lets say you go out on an outward journey of 5km and lets say that run lengths are uniformly distributed between 0 and 10km (so that the mean is 5km, your run length). If you observe someone running for 2 km then you can eliminate those in the 0-2km part of the distribution and so I think you can say the mean run length of that group is 6km. If you see someone for the whole 5km then they are in a group with a mean run length of 7.5km. But the results is highly sensitive to the distribution you assume and where your run length falls in that distribution.

2. Originally Posted by Rincewind
I'm not sure but I suspect it will greatly depend on what you assume about the distribution of possible run lengths. Also it probably depends on how long you observe them for. Obviously if you observer for the whole of your ourward journey and haven't turned when you do turn then they are definitely running further than you.

To make calculations easier lets just talk about the outward journey and say the run finishes when you turn. Lets say you go out on an outward journey of 5km and lets say that run lengths are uniformly distributed between 0 and 10km (so that the mean is 5km, your run length). If you observe someone running for 2 km then you can eliminate those in the 0-2km part of the distribution and so I think you can say the mean run length of that group is 6km. If you see someone for the whole 5km then they are in a group with a mean run length of 7.5km. But the results is highly sensitive to the distribution you assume and where your run length falls in that distribution.
Thanks. Yeah I was thinking along the lines that if I see someone do 2k and don't see them turn then they are doing at least 2k, but if I see them do 2k and then turn then they must be doing at least 4k.

I don't know what the distribution would be, I would assume it's something like the second half of a bell curve.

3. Originally Posted by road runner
I don't know what the distribution would be, I would assume it's something like the second half of a bell curve.
You can say some general things without knowing too much about the distribution. Firstly the distribution of very short runs is more important than the details of the distribution of longer runs. Since the calculation is basically about reevaluating the mean after excluding the part of the distribution which doesn't apply due to the length that you have already observed. For example if no-one runs for less than 2 km so that run lengths are uniformly distributed from 2-10km (mean of 6km) and you see someone run for 1 km then you haven't got any reason to update your initial guess of 6km. Secondly if you assume your run length is the mean of the total distribution then anyone you observe you would initially give a guess of them running for the same distance as you. As you observe them over a longer distance you might adjust this up but you would never adjust it down unless you saw them both start AND end, in which case you would know their run length exactly.

4. Originally Posted by Rincewind
You can say some general things without knowing too much about the distribution. Firstly the distribution of very short runs is more important than the details of the distribution of longer runs. Since the calculation is basically about reevaluating the mean after excluding the part of the distribution which doesn't apply due to the length that you have already observed. For example if no-one runs for less than 2 km so that run lengths are uniformly distributed from 2-10km (mean of 6km) and you see someone run for 1 km then you haven't got any reason to update your initial guess of 6km. Secondly if you assume your run length is the mean of the total distribution then anyone you observe you would initially give a guess of them running for the same distance as you. As you observe them over a longer distance you might adjust this up but you would never adjust it down unless you saw them both start AND end, in which case you would know their run length exactly.
Yeah ok, thanks

5. I think this is actually a very interesting problem set. Another variable that probably needs defining is what sorts of runs the people are going on. For instance if the runners are all running from random points in a large grid at random times, then I would argue that seeing someone running at all skews the probabilities well above the mean and towards a longer run, since if that person had done a shorter run there is a higher probability of you not seeing them at any point of it at all. On the other hand if it's a linear coastal road and everyone runs within a certain area and time window such that the runners tend to overlap and see each other at some stage then that issue could be less significant.

The turning around bit is interesting because it automatically doubles the minimum distance they could be running for among the range that was previously possible. However if someone turns around really quickly, they become more likely to be on a shortish run, because the probability of seeing the turn in such a small distance when intercepting a random section of run is much higher for a shorter run than a long one. This is only true up to a point though - eg if we see someone run 990 metres then turn, their total run is now less likely to be, say, 2 km long (since if it was we would be more likely to have seen them turn faster).

Once we see someone turn around we get no new information about the length of their run until they get back where they were, which if I'm not mistaken means that seeing someone turn around has the same immediate impact on the estimate as seeing someone finish.

6. Yes and another factor worth considering the is speed. I would imagine the faster someone runs you could infer that their total run distance to be more likely to be shorter than longer.

I thought about the the geometry and running to and fro but basically decided to just thing about a linear situation with running going in one direction because it is the easiest to analyse and it is worth see what you could say about that before complicating the problem.

7. Originally Posted by Kevin Bonham
I think this is actually a very interesting problem set. Another variable that probably needs defining is what sorts of runs the people are going on. For instance if the runners are all running from random points in a large grid at random times, then I would argue that seeing someone running at all skews the probabilities well above the mean and towards a longer run, since if that person had done a shorter run there is a higher probability of you not seeing them at any point of it at all. On the other hand if it's a linear coastal road and everyone runs within a certain area and time window such that the runners tend to overlap and see each other at some stage then that issue could be less significant.
It is a coastal path, but I don't live on the coast so I run about 3k before I get to the path. Some of the other runners might drive and park there, or some might do what I do but live much closer or possibly even further away.

The turning around bit is interesting because it automatically doubles the minimum distance they could be running for among the range that was previously possible. However if someone turns around really quickly, they become more likely to be on a shortish run, because the probability of seeing the turn in such a small distance when intercepting a random section of run is much higher for a shorter run than a long one. This is only true up to a point though - eg if we see someone run 990 metres then turn, their total run is now less likely to be, say, 2 km long (since if it was we would be more likely to have seen them turn faster).
Yes what I was thinking was that if someone was doing 20k and I see 2k of it, then I have only about a 1 in 5 chance of seeing them turn. But if I observe the same thing with someone doing 5k, then my chance of observing them doing 2k including a turn is much higher. So I was thinking if I see that, then it's more likely to be someone doing a shorter distance.

Once we see someone turn around we get no new information about the length of their run until they get back where they were, which if I'm not mistaken means that seeing someone turn around has the same immediate impact on the estimate as seeing someone finish.
Yes you might be right

8. Originally Posted by Rincewind
Yes and another factor worth considering the is speed. I would imagine the faster someone runs you could infer that their total run distance to be more likely to be shorter than longer.

I thought about the the geometry and running to and fro but basically decided to just thing about a linear situation with running going in one direction because it is the easiest to analyse and it is worth see what you could say about that before complicating the problem.
You're right people definitely do on average run faster for their shorter runs but I think there's a few complicating factors:

1. If someone is running much faster or slower than I am, then I'm not going to see them run very far because they will either disappear around a bend or I will pass them. I'm going to make the most useful observations of runners of about my pace.

2. Runners have different inherent paces, so while Runner 1 might do short/medium/long runs in a pace of say 3/4/5 min/km, runner 2 might be 5/6/7 min/km. If I see someone doing 5 min/km how will I know if that's their fast or slow speed?

3. Building on 2, the distances that comprise short/medium/long will also vary a lot too. So runner 1 might do say 5/15/30 km runs, Runner 2 might do say 3/5/10.

9. If I ignore the factor of missing someone completely by not seeing any of their run, then if I see someone turn and know nothing about the distribution, I'm going to assume they're running an average of four times the distance that I've seen them run. However if it's a situation where I'm likely to miss the short runs entirely, I'll increase that estimate. Also if I know the distribution and know that that would put them at one end of it or outside it, then I'll adjust that multiplier down or up accordingly.

10. In the following situation:

* randomly-timed runs on a large grid where the chance of seeing a given runner during their run at all is proportional to how long they are running for (in practice this is never quite true as you might see them start their run from a distance)
* distance of runs is randomly distributed between 0 and n km, and is not necessarily an integer but can be any number.
* assumed "stalker factor" such that once you see someone run you follow their path and keep seeing them
* no turnbacks, just one way runs
* all runners run at same speed and look equally fit and/or tired

... I get that on seeing someone running at all I will assume they are running an average of 2n/3 km.
For any distance then run I then increase that estimate by a third of that distance.

(I haven't proven this, just derived it from simulations. It is probably easy to prove if correct.)

For the same case but where there is a minimum distance per runner greater than 0, I get that even seeing someone run less than the minimum distance (provided they are still running) still justifies an increase in the estimate while they are still running. The reason for this is that the probability of seeing a runner run a given distance (as opposed to just running at all) decreases more sharply with distance if their overall distance is small than if it is large. If the distance range is 2 to 10 km then I have five times as much chance of seeing the 10 km runner running at all, but nine times as much chance of seeing the 10 km runner run for at least one kilometre.

I haven't worked out how to even approximate probabilities for turnbacks (or for seeing someone running who then ends their run) yet.

Suspect I would have known how to derive a lot of this stuff exactly in my late teens but have forgotten much too much now.

11. RIP Maryam Mirzakhani, first and so far only woman to win the Fields Medal, dead at only 40.

12. Originally Posted by Kevin Bonham
RIP Maryam Mirzakhani, first and so far only woman to win the Fields Medal, dead at only 40.
Very sad. As Kevin notes only female Field's Medalist so far.

13. I simulated a turnback version of the runner problem as well using random simulations. So for the following parameters:

* probability of seeing runner run at all is proportional to distance run
* runner is followed once seen
* runner runs a random distance between 0 and n km, turning back halfway
* runner has been followed for d km at the point of making an estimate

... I get that if a runner is seen turning back after d km, the average total distance of their run is on average simply d+(n/2). So if the run is 0-10 km, then seeing the runner turn back, you just estimate their total run distance is 5 km plus what you've already seen.

I haven't worked out how to simulate a runner not turning back in this case but as they approach n/2 without turning back their total distance run will obviously approach n. (Indeed if they get to n/2 they are running n and will turn back immediately).

Given that at the moment of seeing a runner (ie d=0), the average distance they are running is 2n/3 with or without turnbacks, I suspect that seeing a runner turn back will always decrease the estimate of how far they are running, unless they are running exactly the maximum and you've seen them run exactly half of it. I wouldn't be surprised if the formula turned out to be (2n+2d)/3 for a runner who has not yet been seen to turn back, but it might be more complicated.

14. Thanks KB

15. Not sure if Rincewind and others here would have seen this one before but the (well known) result is rather nice. A fellow electoral expert told me about it today.

Voting papers for an election between two candidates are counted one at a time. Candidate A ends up winning the election with p votes, while Candidate B receives q votes and loses. What is the probability that candidate A will lead throughout the count (no ties once at least one paper has been counted)?

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