This doesn’t really say a lot unless it factors in frequency somehow, maybe the line alpha is number of journeys per day or something. Not of people, I mean if availability. In the U.K. I can see lines in areas that the reality is there’s one a day, which is almost useless, yet this puts the route on a par with others that may have 2 an hour.

WWW is a famous example where it’s actually less syllables to pronounce the full phrase - World Wide Web.

Finally got round to knocking up a simulation, can confirm that the two methods (randomly choosing whether to bet H or T, or using the lagged counts) both give the same (0.5) win rate, as expected. Indeed, any strategy (e.g. always choosing H or T) will yield the same win rate for a truly random coin. I suspect there was a slight glitch in your friend’s simulation. Happy to share (R) code and plot(s) if you want.

They should be random enough, albeit improperly generated random numbers could be a cause. My money is on them setting up the simulation incorrectly, though. When I say incorrectly, I mean it could be that they haven’t set the simulation up quite as they think they have.

Ah! If I were you I would stick with your original thinking as I’m reasonably confident there was a coding issue or the set-up of the simulation wasn’t quite what it should be. If I have time I might try and knock something together myself, but that’s a big if as I’ve got a mental couple of weeks.

I’m with you. I think if your friend’s code showed otherwise then there’s a flaw in the code and it’s not doing quite what it should be. Would they be willing to share it?

>I actually was debating this with a friend so vigorously that he coded a program to compare yhe wins of betting randomly or on lagging possible outcomes of a randomly generated number. Betting on lagging outcomes brought more wins.

Can you elaborate more on what you mean? I think I misunderstand you somehow as it seems to me betting on lagging outcomes cannot outperform betting randomly in the long run or the outcome is not random (or there’s a coding / concept error).

>Professional sports is not quite random though.

Absolutely.

The latter is because of the former.

Even more recently MS have provided their own solution for Windows 10 (provided you’re running a sufficiently updated version) and 11, which supports both X11 and Wayland.

See here albeit this document is slightly out of date because, as of just a few days ago, they now support Windows 10 from 19.044 (21H2) onwards iirc. Edit: as per here.