Original rules (host knows where car is and always opens a door with a goat):
- 1/3 of the time your original choice is the car, and you should stick
- 2/3 of the time your original choice is a goat, and you should switch
Alternative rules (host doesn't know where car is, and may open either the door with the car or a door with a goat)
- 1/3 of the time your original choice is the car, the host opens a door with a goat, and you should stick
- 1/3 of the time your original choice is a goat, the host opens a door with a goat, and you should switch
- 1/3 of the time your original choice is a goat, the host opens the door with the car, and you're going to lose whether you stick or switch
So even under the new rules, you still only win 1/3 of the time by consistently sticking. You're just no longer guaranteed that you can win in any given game.
Well yes, if you throw out half of the instances where your original choice was wrong, then the chance your original choice was correct will inevitably go up.
> the logic of Monty Hall Doors does not make sense to me
For me, the core is that you have a 1 in 3 chance of getting it right on your first guess, and nothing can change that. So if you always stick with your original guess, you will win one third of the time.
No no. The thing is, the Monty Hall guy is responding to YOUR choice. So if he has to open a door where you fail, it's a response to what he knows of your choice, so HE knows what YOU chose and is not only revealing the remaining losing choice but also the winning choice. Call it a coin flip except for he always has to call tails.
Therefore your choice can either be cadillac or goat, he cannot choose cadillac and has to show a goat, so the remaining option you DIDN'T highlight is that much more likely to be cadillac because it could've been either, but he doesn't get to pick randomly, he had to show which one was NOT the winning one.
Hence the result. And since it started out as one pick of three, he responds to you and then you respond to the added information by switching and that's where the 66% odds come from: two moves each responding to each other.
Your explanation isn't wrong, but it's never quite resonated with me because it feels almost like a magic trick than something that follows intuition. Like seeing a magician perform a trick, it doesn't quite convey to me the "why" as much as the "what", and even though I know there's no actual magic, I still feel like I'm left having to figure out what happened on my own.
The idea that finally made it click for me is that Monty has to choose one of the doors to open, and because he knows which door has which thing behind it, he'll never pick the door with the winning prize. That means the fact that he didn't pick the other door is potentially meaningful; unless I picked the right door on my first try, it's guaranteed to be the one he didn't open, because he never opens the right door on his own. His choice communicates meaningful information to me because it's not random, and that part while seemingly obvious gets left implicit in almost every attempt to explain this that I've seen.
Another intuitive way to explain it would be to imagine that the step of opening one door is removed, and instead you're given the option of either sticking with your original door or swapping to all of the other doors and winning if it's any of them. It's much more obvious that it would be a better strategy to swap, and then if you add back the step where he happens to open all of the other doors that aren't what you picked or the right one, it shouldn't change the odds if you're picking all of the other doors. This clarifies why the 100 door case makes it an even better strategy to switch than the 3 door case; you're picking 99 doors and betting that it's behind one. The way people usually describe that formulation still often doesn't seem to explicitly talk about why the sleight of hand that opening 98 of the doors is a red herring though; people always seem to state it as if it's self-evident, and I feel like that misses the whole point of why this is unintuitive in the first place in favor of explaining in a way that clarifies little and only makes sense if you already understand in the first place.
Originally, a font (also spelled fount, at least formerly) was a physical thing: a collection of metal slugs, each bearing the reversed shape of a letter or other symbol (a glyph, in typographical parlance). You would arrange these slugs in a wooden frame, apply a layer of ink to them, and press them against a sheet of paper.
The typeface dictated the shapes of those glyphs. So you could own a font of Caslon's English Roman typeface, for example. If you wanted to print text in different sizes, you would need multiple fonts. If you wanted to print in italic as well as roman (upright), you would need another font for that, too.
As there was a finite number of slugs available, what text you could print on a single sheet was also constrained to an extent by your font(s). Modern Welsh, for example, has no letter "k": yet mediaeval Welsh used it liberally. The change came when the Bible was first printed in Welsh: the only fonts available were made for English, and didn't have enough k's. So the publisher made the decision to use c for k, and an orthographical rule was born.
Digital typography, of course, has none of those constraints: digital text can be made larger or smaller, or heavier or lighter, or slanted or not, by directly manipulating the glyph shapes; and you're not going to run out of a particular letter.
So that raises the question: what is a font in digital terms?
There appear to be two schools of thought:
1. A font is a typeface at a particular size and in a particular weight etc. So Times New Roman is a typeface, but 12pt bold italic Times New Roman is a font. This attempts to draw parallels with the physical constraints of a moveable-type font.
2. A font is, as it always was, the instantiation of a typeface. In digital terms, this means a font file: a .ttf or .otf or whatever. This may seem like a meaningless distinction, but consider: you can get different qualities of font files for the same typeface. A professional, paid-for font will (or should, at least) offer better kerning and spacing rules, better glyph coverage, etc. And if you want your text italic or bold, or particularly small or particularly large (display text), your software can almost certainly just digitally transform the shapes in your free/cheap, all-purpose font, But you will get better results with a font that has been specifically designed to be small or italic or whatever: text used for small captions, for example, is more legible with a larger x-height and less variation in stroke width than that used for body text. Adobe offers 65 separate fonts for its Minion typeface, in different combinations of italic/roman, weight (regular/medium/semibold/bold), width (regular/condensed) and size (caption/body/subhead/display).
The P.G. Wodehouse translation?
reply