There was a post a bit back (which you might be referencing) about almost this exact situation, except it was about black hat SEO dramatically backfiring. Specifically, businesses were asking bloggers and news sites to delete spam comments that linked back to the sites.
The HTML that Mahbod wants you to paste looks pretty similar to what those spam comments were like...
> There was a post a bit back (which you might be referencing) about almost this exact situation, except it was about black hat SEO dramatically backfiring. Specifically, businesses were asking bloggers and news sites to delete spam comments that linked back to the sites.
One approach that really helped me out was to extend the problem from three doors to a million doors.
You're allowed to pick one door, out of a million. Then, the host opens 999,998 of the other doors which he knows have a goat. This leaves the closed door you picked and one remaining closed door.
So, there's now two possibilities. Out of one million doors you somehow just happened to have guessed right and picked the right one. Or, you're wrong. If you're wrong then the correct door would be the other closed one.
I do the same thing by expanding it out to more doors, but when talking about it or explaining it to others (great cocktail party discussion btw), I use 100 doors, to make the percentages easy to follow.
Example - there are 100 doors, you pick 1 that has the grand prize. There's a 1% chance you picked the prize on your selection. Monty Hall eliminates 98 of the doors, meaning there are 2 left. There's STILL a 1% chance you picked the door with the prize, but that means there's a 99% chance the only remaining door has the prize. Obviously, you should switch.
Another way is to take a step back and forget about revealing the goats. You have your initial selection (using your numbers) with a 1% chance of picking the prize. Now you're presented the option to leave it shut and open all other doors keeping the prize if it's in any one of them. So if the prize has equal odds of being in any given door, then opening 99 of them gives you a 99% chance of finding the prize versus the 1% chance of only opening one door. This has helped some people I've talked to realize that the odds the prize is in that remaining door (in the proper Monty Hall problem) is actually the sum of the odds it's in all the doors you didn't select.
EDIT: I've also demo'd this using 3 cards (an ace and two others), but I like the idea of demonstrating with a larger scope. Call the ace of spades the prize, let the player select one card from the deck at random. Now they can keep that card or scour the remaining 51 cards for the ace of spades. A good tactile demonstration that they have a 1/52 chance (~2%) of having selected correctly the first time versus ~98% that they were wrong.
Consider the 3 door case, it's narrowed down to 2 doors. The one you selected has a 1/3 chance of being correct, the one you didn't has a 2/3 chance. If you randomly select between the two (50/50 odds) you come out with a 50% chance of picking the correct door. If you strictly switch, your odds are better, and if you stay the same the odds are worse.
Random choice would be optimal if the host didn't know the location of the car. If the host is blind & car wasn't revealed already, there's 1/2 chance. If the host knows & the car is hidden, then the remaining door has 2/3 chance
The way I explain is to expand it but also change the frame of reference to so its clear that the host is an adversary. Imagine we play a game called "who has the Ace of Diamonds"? I deal you one card face down and I deal me 51 cards. I look at my cards and then choose 50 of them to show you, none of which are the Ace of Diamonds. Do you want to keep your card or take the one I have not turned over?
The fact that you asked this question shows what's wrong with the supposed "problem". In the usual presentation, it's a "trick question" because the person presenting it omits the crucial information that the offeror is intending to act adversely to the decsion making subject.
Without that information, the interlocutor naturally assumes that the offeror opens doors randomly - in which case the fact that he knows what's behind each door is an irrelevant "red herring" and the correct conclusion is different.
In the random-opening case the intuitive conclusion is right - that the agent (offeree)'s decision does not affect the odds. But if the question-poser revealed that the offeror is trying to maneuver the offeree into the result that's less valuable for the offeree, then the supposed mystery of the whole thing evaporates and the correct conclsion is obvious.
Ha, interesting. Yes, well, back when the problem was invented it was assumed that everyone was aware of the TV show "Let's Make a Deal" and how the scenario worked. The host/offeree (a man named Monty Hall) _never_ picked a door that had the prize behind it, implying that he _always_ knew where the prize was.
It does. If the host doesn't know, there's a 1/3 probability that the game will end (you won't get a chance to switch) if the host picks the car. If this doesn't happen, it's 50:50 whether you keep your door or switch to the remaining unopened door.
Let's say that you have 3 doors and you pick door #1.
At that point the host says "Ok, I'm going to open one of the other doors at random and then give you the chance to change your choice".
At this point 2 things could happen.
Either he opens the door with the car in which case you will definitely switch to that door and have a 100% chance of winning.
Or he opens the door with a goat.
There is a 1/3 chance that you picked correctly in the first place, therefore there is a 2/3 chance that you picked wrongly, so it's safer to assume that you picked wrongly.
You know that he didn't show you the car, so either you picked correctly to being with (1/3) or you picked wrongly (2/3), but if you picked wrongly then it must be behind the door that the host did not open. So you still have better odds from switching even though the host did not know.
You are correct, as long as the host doesn't reveal the prize the odds of winning by switching remain better than the odds of winning with your initial selection. EDIT: And in the case that he reveals the prize your odds have gone up. I suspect (but haven't calculated) that a version where you can switch to the winning prize when it's revealed would give you much better odds of success and a bankrupt game show would result.
You're right. When doors are opened randomly you can not tell whether the host was "lucky" (to not reveal the car) or whether you picked the right door to begin with since these are both equally likely.
Let's assume that both of these game shows are showing on different channels and you were invited to participate in one of them. But once you get to the studio you have a case of nerves and forget which show you are on. Should that change how you make your decision?
No, the odds for switching at the real show are 66-33 and the odds at the fake show are 50-50, you might as well switch after the reveal (what you are calling 'better odds' boils down to 'sometimes you can see the car so you switch to it'. When you see a goat, you have equal information for the remaining doors.).
I will never understand how one million doors is easier to understand than 3. Switching doors means you get two doors out of three doors. 66% > 33%. 66/33 illustrates the benefit much better than 1 vs 2 out of a million.
I think what confuses people is that they think a random door is revealed, whereas the problem says a goat is specifically chosen before the offer to switch.
This is definitely not usable as an early intro to CS book, as it's pretty dense. There's a wide variety of introductory material that you can pick from, depending on your interests (what kind of stuff are you interested in doing). But TAOCP is aimed at someone a bit more advanced. Since you have a math background though I think you could give it a shot and see how far you get.
I feel like a lot of this trendy stuff comes out because it demos very well. The designer shows it off in a controlled setting, everyone's wowed and impressed and glad they hired a "good" designer, and nobody bothers to check whether it's actually useful at all. Someone should've pointed out the website is there to sell the product, not to showcase the FE guys's skills.