3/4 + 3/16 + 3/64 + 3/256 + ... is easy to visualize as successively filling in three quarters of an ever-smaller residual square. Intuitively, no matter how finely you detail it, you're never going to stop fitting inside the original area-1 square.
Now squeeze the top-left piece so it's half as wide but twice as tall (i.e. the same height as the full rectangle), and do this recursively. Same area, right? Then stack the L shapes on top of each other. Then you have one of these horns.
Huh? The horn is a three-dimensional object; the square exists in 2-space. You can't make the horn from pieces of the square even if you allow deformation.
The best way to think of this is that there are no odds at all. All money goes into a pool and is shared evenly amongst the winners. That's it. The fewer people* that bet on the winner, the larger their share.
Talking about the odds while the betting is open is really talking about what the payout would be if the event occurred with the current pool.
*It is of course about the amount of money, not the number of people.
That's a good question. They may not state it explicitly but all betting sites would have an escrow (or 'wallet') system. The alternative, where if you lose, they send you an invoice for what you owe would be, er, uncommercial.
I think it's a worthwhile exercise to look at situations like this and objectively asses the indiscretion. My first response was, 'that's fine, he's just talking about his girlfriend, no issue'. This is an oversimplification. The offence comes from the humour, which relies on the implication that all girlfriends are like this. It's a little subtle but it's there.
I don't really like males (I'm male) being portrayed as domestic buffoons who can't cook, are too lazy to clean, even in the context of light humour. It's a false stereotype.
Your links are about comedians getting into trouble by offending people. That is a different issue to gender stereotyping in a professional environment. People do have a right to be offended, call people out on it, and try to improve the world they live in. I don't think that makes then sad.
Apologies are by definition when you apologies for your own actions. To claim that it applies to people based on their reaction or is explicitly for people who reacted a certain way is not IMHO a real apology. This type of conditional apology (called the 'Carey' apology where I'm from http://www.heraldsun.com.au/news/my-vile-apology/story-e6frf...) is an apology for the reaction in others, not the act itself.
The spacial resolution of analog photographs is not at the scale of individual molecules. It is limited by the "grain" of the film [1]. Other factors include distortions from lens shape imperfections in the lens material.
Audio tapes record their signal in magnetic domains. These are areas of tape the are large enough to be written and read. Much larger than anything approaching the scale of individual molecules.
Sound recordings on vinly are similarly limited by other factors. To "read" the signal from a record groove a needle is dragged a needle through a v shaped groove. This needle jiggles up and down and left and right, hence the two channels in stereo. The time resolution of the signal (max frequency) is limited by the physics of the jiggling head. It's inertia prevents it from reading signals over a certain limiting frequency. Again, well before the molecular level.
A couple points to add to that: with vinyl, the recording process itself (for which there are several techniques) also adds noise. The recording head (in whatever form) is also subject to physical laws. The materials used at each stage of the process are also not ideal substrates. Hell, vinyl is a polymer, which means it's a big tangle of chained molecules. Even if everything else were perfect, it would not be possible to write data at a molecular scale into such a medium.
I figured my layman explanation would suffice. While we're in the realm of correct details, it's also worth nothing that with each use of a needle dragged through a groove (or even with magnetic tape dragging across all the bits in the tape machine) the surface is degraded and the original recording along with it. Less fidelity, more noise ... hiss, pop, crack
Digital gives us error correction so that even when the physical media starts to degrade (taking the signal with it) our firmware and our software work out where the mistakes exist and attempt to correct them. Of course, once the signal has degraded beyond the point of recoverability, you "suddenly" have a catastrophic failure. (And this is why backups are important...)
