6 ms·

Nice article. It reminds me of my year living in London, and taking the bus everyday to Imperial College from West End Lane in West Hampstead. There was a stop

Nice article. It reminds me of my year living in London, and taking the bus everyday to Imperial College from West End Lane in West Hampstead. There was a stop on both sides of the road - one for the outbound bus, and one for the inbound (the bus went from central London to a terminus and then returned mostly on the same route). Now we did not use schedules - way too inaccurate at rush hour, and the busses there were pretty frequent anyway. But we did expect an even chance of the inbound bus arriving before an outbound one did. My daughter and I became convinced after a while that this was not happening, so we invented a game (which we called "The Game of Life".) When our bus (inbound) arrived first, we added 1 to our score. We subtracted 1 for every outbound bus that passed before ours arrived (there were often more than 1). We realized that the result would be slightly skewed to the negative, but we expected the outcome to be close to 0 over time. Of course it was not. Anyway we extended the game to many statistical situations. For example, you go to the checkout line at the supermarket, and there are N people in front of you. When you get to the front of the line, you count the people behind you - call that M. If M is bigger than N, you scored life points. If it is smaller, you lost some. So you add M-N to your running score, and you get an idea of how lucky you are in life. However, I never followed up with any real analysis, so I enjoyed this article.

asdkhadsj 8y ago
Wow, interesting idea! Imagine if (in the future) some item like a phone can detect this information around you, and automatically record it. Forming games ontop of this life data would be weird, neat, fun and sad all at the same time. Imagine seeing a real example of where someone else is just more lucky than you are in stupid but impactful (on your morale) ways. If it didn't seem so tedious to track, I'd love to implement an app to record this info. Unfortunately no one I know would care, and I'm sure I'd get too lazy to keep it accurate. Neat nonetheless, thanks for the cool thoughts :)
- adrianN 8y ago
  Similar to the recruiters that throw away the top half of the application stack because they don't want unlucky people in their company I could see such data become valuable to some people.
bostonpete 8y ago
Your use of "of course" seems to imply that there's some statistical reason that the probability of the next bus being inbound vs outbound wouldn't be equal. Is there? If so, it seems like it must be a different reason than the one in the article. What am I missing...?
- theoh 8y ago
  It is impossible to catch more than one inbound bus on any given occasion, whereas any number of outbound buses might pass. BTW, on a slightly unrelated point, if there's no timetable, but the interval between buses is maintained reliably, the expected waiting time is uniformly distributed over that interval. If you have to get a second bus, you need to convolve two of those two uniform distributions to find out the distribution of overall journey times. This is a trapezoidal distribution, which is just about analytically manageable. But a journey with two transfers (3 buses in total) results in a likely overall time distributed according to a uniform distribution convolved with a trapezoidal distribution, which is a very weird non-smooth shape. You can see why people choose to model distributions with Gaussians, which are well-behaved (convolve two Gaussians, get another Gaussian). The Gaussian just lends itself ideally to recursive applications, hence recursive filtering (e.g. Kalman filters).
  - LolWolf 8y ago
    Also, gaussians are great approximations for large n, too, since the convolution of any distribution with itself n times (for n "large enough") is close to gaussian (by the CLT. More generally, there are very nice error estimates for many distributions). I suspect this analysis can be carried out and yield quite good results in the gaussian case (a careful analysis might even yield error bounds on the result).
    - theoh 8y ago
      Yes. If you spend your whole life on one long multi-transfer bus journey, you'll end up with a gaussian. It's a bit less clear that gaussians should be used when e.g. fitting a coordinate to an astronomical feature, which might not actually be symmetrical. The other useful property that the gaussian has is its separability, in the 2D case. That is unique to the gaussian and counts for a lot.
hanoz 8y ago
This is pretty counterintuitive. In the game described you should expect to see one 'wrong way' bus per play on average, not half as you might expect. On the other hand, you have an exactly even chance of catching your own bus before seeing a wrong one, so if your scoring system had been +1 for your bus and -1 for one or more wrong ones, then you would indeed score 0 over time. But with your point per bus scoring system your expected score turns out to be -0.5 per play.

asdkhadsj 8y ago

Wow, interesting idea! Imagine if (in the future) some item like a phone can detect this information around you, and automatically record it. Forming games ontop of this life data would be weird, neat, fun and sad all at the same time. Imagine seeing a real example of where someone else is just more lucky than you are in stupid but impactful (on your morale) ways. If it didn't seem so tedious to track, I'd love to implement an app to record this info. Unfortunately no one I know would care, and I'm sure I'd get too lazy to keep it accurate. Neat nonetheless, thanks for the cool thoughts :)

adrianN 8y ago
Similar to the recruiters that throw away the top half of the application stack because they don't want unlucky people in their company I could see such data become valuable to some people.

bostonpete 8y ago

Your use of "of course" seems to imply that there's some statistical reason that the probability of the next bus being inbound vs outbound wouldn't be equal. Is there? If so, it seems like it must be a different reason than the one in the article. What am I missing...?

theoh 8y ago
It is impossible to catch more than one inbound bus on any given occasion, whereas any number of outbound buses might pass. BTW, on a slightly unrelated point, if there's no timetable, but the interval between buses is maintained reliably, the expected waiting time is uniformly distributed over that interval. If you have to get a second bus, you need to convolve two of those two uniform distributions to find out the distribution of overall journey times. This is a trapezoidal distribution, which is just about analytically manageable. But a journey with two transfers (3 buses in total) results in a likely overall time distributed according to a uniform distribution convolved with a trapezoidal distribution, which is a very weird non-smooth shape. You can see why people choose to model distributions with Gaussians, which are well-behaved (convolve two Gaussians, get another Gaussian). The Gaussian just lends itself ideally to recursive applications, hence recursive filtering (e.g. Kalman filters).
- LolWolf 8y ago
  Also, gaussians are great approximations for large n, too, since the convolution of any distribution with itself n times (for n "large enough") is close to gaussian (by the CLT. More generally, there are very nice error estimates for many distributions). I suspect this analysis can be carried out and yield quite good results in the gaussian case (a careful analysis might even yield error bounds on the result).
  - theoh 8y ago
    Yes. If you spend your whole life on one long multi-transfer bus journey, you'll end up with a gaussian. It's a bit less clear that gaussians should be used when e.g. fitting a coordinate to an astronomical feature, which might not actually be symmetrical. The other useful property that the gaussian has is its separability, in the 2D case. That is unique to the gaussian and counts for a lot.

hanoz 8y ago

This is pretty counterintuitive. In the game described you should expect to see one 'wrong way' bus per play on average, not half as you might expect. On the other hand, you have an exactly even chance of catching your own bus before seeing a wrong one, so if your scoring system had been +1 for your bus and -1 for one or more wrong ones, then you would indeed score 0 over time. But with your point per bus scoring system your expected score turns out to be -0.5 per play.