Counter-Intuitive Science

Yes, at the very beginning, when no door has been chosen nor opened. You have a 1/3 chance of winning the car (if the presentator will open one at random)However, when the presentator opens a goat, you know you are not in sit. C1 and C2, and so you know your chances just went up to 50/50 It also eliminates one of A1 and A2 (in fact, they become one and the same)Without this, there would be A1,A2,B1,B2 which would mean, even with three doors, that you would be randomnly picking the car half of the time. (which we know is not true, since you will be picking it 1/3 of the time)

Then How would you formulate it then ?I get what you mean, but I thought maybe the easiest way would be to just see it from the referential of the contestant if you will. From the eyes of the contestant, the two situations are identical. You choose a door, he opens one of the two left and reveals a goat, and he asks you if you want to change ?Now it is just to realize that if someone from behind a curtain behind you would say: ''He deliberately opened the goat, he knows where the car is'' or ''he opened a door randomnly'', then you should then know that the probabilities aren't distributed in the same way in both situations.But if there is a better way to present it, I'm open Ha, didn't know that expression. And know, it would prove difficult to appear unreasonably stuborn about a point with me. Remember, I study Math and Physics, so the mathmatician side of me knows the devil is in the details

2/3 ? The line can be drawn anywhere in the circle ? I mean, it's length can range from 0 up to D (diameter) ?AbE I get 25% (1/4)Edited by slevesque, : No reason given.

Anyone else can solve these problems, see if I'm at least half-right ?I have a feeling Noetherian Atheist won't be coming back to give me the answers

It seems so obvious I tell myself exactly the same thing I had thought about the reasoning for 1/2.But here's how I viewed to to get the 1/4. Each line drawn inside the circle will have a middle point. For example, if the middle point is at the center, then the line is actually the diameter. So I calculated the area inside the circle where the middle point can be which would involve a line longer then the side of the triangle. This area is a smaller circle with a diameter of D/2 (D being the diameter of the bigger circle) at the center of the bigger circle. Which has a fourth of the total area of the circle.AbE I don't feel this is clear. It's a circle in a circle, and whenever the middle point of the line is in the inner circle, the line is longer. Everywhere else in the outer circle it is smaller. Since the area of the inner circle takes 25% of the area of the outer circle, so will the line be longer then the sides of the triangle 25% of the time.Edited by slevesque, : No reason given.

The question that is asked is essentially: Out of the three, what is the chance I get a card that has the same color on both sides ?Seen this way, it is obvious that the answer is 2/3.Seen another way, there are 6 faces in total (2 per card). half are green, and half are red. Whichever color you choose, you have 2/3 chances to have picked the card single-colored card, since it has 2 of the total three faces of that color.In other words. Suppose you get red. You either picked the red/red card on it's face no1, or on it's face no2 or the red/green card on it's face no1. Idem for green.

I would guess that it does, or at least that there is a way of finding the correct probability.I remember my physics teacher last year asked us the following math question: given a stick that is a meter long, what is the probability that, by cutting it in 3 parts, I can take those three parts and form triangle.Which is kinda similar to your question, since there are, mathematically, infinitely many ways to cut a stick in three pieces. Yet there was still a real answer.Yet I still do feel quite confident about my 1/4 answer, I have given much thought into it and it still feels like the right approach. While something which I can't put my finger on is missing in the two other reasonings who give 1/2 and 1/3

That all points aren't born equal ? I have been thinking that an infinit number of lines can have there middle-point be at the middle of the circles. While this is not true for any other lines drawn, as their middle points will be unique to them. (at least that's how I see it, my visualisation could be wrong) I couldn't answer those questions. I actually asked the question to ''Dr.Math'' and they sent me back this answer: Which seems to be about what you are saying.I've never had any course in probabilities and statistics, and so when I happen to do problems like these I go with my intuition. This is a case where the reality of the problem was counter-intuitive; randomness isn't just 'randomness', and defining it in different ways will lead to different results all equally valid.But the question still remains for me: how can we know which one is more valid ? If we can extend the survey analogy, there should be only one actual outcome in reality. How can we know which one it will be ?