if it's opened randomly then it doesn't matter, it is 50-50. You have to account for the situation where the host might reveal the prize in his random pick, and by not doing so you learn that your choice is more likely to be the prize than initially thought.
If you picked the prize P(A = 1/3), then the random open is going to be Goat P(1). if you didnt pick the prize P(B = 2/3) then the random open is going to be Goat P(1/2) or Prize P(1/2)
P(Goat reveal) happens (1/3)(1) + (2/3)(1/2) = 2/3 times
P(Prize reveal) happens (1/3)(0) + (2/3)(1/2) = 1/3 times
You don't have to account for the chance of him revealing the prize because he does not reveal the prize, he reveals the goat. So switch because it doubles your odds. Or don't, you're a free person. But it would be foolish.
If he's guaranteed to reveal the goat then he knows which door has the prize. This situation he isn't guaranteed to reveal a goat, he just did it this time by chance.
It doesn't matter if he's guaranteed to reveal the goat or not. He did reveal the goat, that's the scenario that's been described. So given that he did reveal the goat, should you switch? The answer is yes, from the information given, you should switch.
>It doesn't matter if he's guaranteed to reveal the goat or not
Yes, it does. this is the entire point behind the popularity of the Monty Hall problem. It is counter-intuitive and feels like overthinking, but it is truth.
I described the situation pretty clearly (imo). I did not need to think about it much at all, though it did take some thought to describe it clearly.
I can tell you, without much thought, having been trained thoroughly in probability and statistics, it is 50-50 if a door is opened randomly to reveal a Goat. You can switch if it makes you feel better - because it is 50-50 so you wont be hurting yourself.
No, it's not. The point of the problem is that it's better to switch than not given the scenario presented. Whether he knows or not doesn't actually change anything, he does reveal a goat. Whether it was luck or deliberate makes no difference.
Switching doesn't change where the prize is more likely to be. That's not how the Monty Hall Problem works. It's the host giving you information about what's happening behind the scenes, by choosing a door to open and you knowing how the host operates.
Whether you end up switching or not, the other opened door is 2/3 likely to contain the prize and the door you picked originally has a 1/3. Whether you switch or not, doesn't change the odds.
I had this problem beaten to death in my probability and statistics courses in college.
To make the problem much more obviously intuitive, simply imagine a thousand or a million doors instead of 3.. and after you make your pick - the host reveals all doors except one to be "failures". Your choice was very obviously 1 in a million, and because the reveal procedure that followed is conditional - if the "success" was behind ANY of the doors you did not pick then it WILL be the only door left in front of you.
It is only a 50-50 IF the revealed doors were opened randomly. In this case, sometimes the "success" would be revealed by accident - and so IF you do happen to find yourself in a position of only having 2 doors left THEN it was more likely to happen if you had actually chosen the "success". So arriving at that situation at all is the conditional information that alters the probability of your original choice - but in this case from 1/3 to 1/2 instead of 2/3. (this is how the probabilities in "deal or no deal" work btw)
Even knowing the solution, it is a good reminder about the power of information derived from systemic processes. the information leveraged in the processes that lead to circumstances is just as important as the circumstances themselves.
The key general point that bypasses most peoples intuition the first time they hear the problem is that the revealed box (or door, etc) is not chosen at random but by a conditional process. The condition is that the host reveals a "failure". In order to pick a failure, he must leverage knowledge about where the "success" is, else he risks revealing it by mistake. Because knowledge of the "success" is embedded into the decision process, more information about the "success" is revealed to the observer which allows them to improve their position.
That does sound pleasant. I wish I could push notifications into a todo list with 1 button / click. I am usually not bothered or distracted by seeing a notification, but having to choose to delete it or act on it immediately is what gets me. I tend to delete everything and keep the interesting ones in the back of my mind for later ("I should check out my gf's new IG post")
I like notifications, and I try to respond to them intentionally. I open apps intentionally (this was easy enough to learn) and close apps intentionally (this was hard to learn). So I am pretty good about actually going into the app to do what I intended to do in response to the notification, and then closing the app. I can reassess afterwards if I want to spend more time in the app and open it back up, but again I create an intention before opening it that has a stopping point.
Intention is the difference between enjoying some funny short videos while sitting in a waiting room, and losing 1-2 hours of your life a day to doomscrolling. And if I could have a filtered list of notifications in a todo somewhere, I can set aside time to update myself on whats new that I care about but arent super important
i still find games to be great for my mental health, even more so now as an adult than as a kid. they are a place where i can make uninformed decisions based on my gut, and experience no negative consequences to my life. i treat every game like a sandbox. dont get sucked into the idea that anything you build or create, or skills you develop, in the game is meant to last.
For example, in an RPG game when I find a powerful 1-time use item - I use it as soon as I can.
There are many factors to optimal decision making, and optimized results are just 1 factor. Games help me explore trade-offs with quick vs long decisions, side effects of over planning, stress and panic effects on decisions, etc.
It carries over into the rest of my life, even work, where I am better able to manage things, like diminishing returns on my efforts, in ways that are very personal to me and my natural tendencies.
Lowkey I think the current education system has evolved from communities that allowed teachers to beat their kids to teach obedience. that obedience was passed down through generations in those communities establishing a culture of obedience for teachers. The current system, although non-violent, is still heavily leveraging the culture of obedience from students. when a kid is disobedient, usually the above mentioned communities threaten to move the kid into an environment where the obedience culture is even more strict as a result of even more violent punishment historically. Early days Public school teachers might lightly smack you with a ruler, early catholic school teachers might spank you, and early military school teachers might straight up beat the shit out of you.
So I think by extending the current system to areas in poverty that do not have an established culture of obedience means we have to either change the system or start beating kids again imo. I vote for the former, obviously.
