Famous brain teaser

Nope I dont get it
Try this, if instead of the host opening a door, what if he asked you if you want to open both the other two doors and if either of them have the car then you win. You'd be getting to choose two doors, not just one. That's obviously better right? Then you have a 2/3 chance of winning if you switch.

That's effectively the same as him opening an incorrect door, he's letting you choose take two chances for the price of one.

Take another look at the million doors example on page 1/2, it really helped me get it.
 
Try this, if instead of the host opening a door, what if he asked you if you want to open both the other two doors and if either of them have the car then you win. You'd be getting to choose two doors, not just one. That's obviously better right? Then you have a 2/3 chance of winning if you switch.

That's how one of my friends explained it intuitively. She saw the answer straight away and pointed at the diagram and said 'that way you get two goes'.
 
1779294515940.gif
 
I like doing it with a million doors.

When you pick your door you have a 1 in a million chance that you've got the car. And a 999,999/1,000,000 chance it is one of the other doors.

You pick door number #327. At this point you can be virtually certain you don't have the right door.

The host then opens every other door except for door #711 showing the booby prize.

If you got your 1 in a million guess right then behind door number #327 is your new car. If you didn't guess right the first time, which let's face it you didn't, then it must be behind the one door the host deliberately didn't open.

This is a famous puzzle, mostly because a lot of people just don't get it. Including a lot of very clever mathematicians.
But with 3 doors you might have picked the correct door to start with, with a million doors you wouldn't have....

This is doing my head in.
 
But with 3 doors you might have picked the correct door to start with, with a million doors you wouldn't have....

This is doing my head in.

I think that's the point. With a million doors, there is no chance that you picked the right door. The host knows where the car is. He opens all the other 999,998 doors. Until there are only two doors left. Your original choice and one other. At that point, what would you do if you were asked to swap.
 
I think that's the point. With a million doors, there is no chance that you picked the right door. The host knows where the car is. He opens all the other 999,998 doors. Until there are only two doors left. Your original choice and one other. At that point, what would you do if you were asked to swap.
Swap
 
The probabilities always have to add up to 1, so you start with 3 doors, which means that each door has a probability of ⅓ of concealing the prize.

1779327922326.png


When you pick a door, nothing can change the probabilities, so your door is ⅓, and each of the other doors is ⅓. All the doors in the group of the doors you chose add up to ⅓, and all the doors in the group of the doors you did not choose add up to ⅔.

1779328334883.png


When the host opens one of the doors in the 2-door group, the probability for that group cannot change, it remains ⅔, but the door he opens has clearly got a zero chance of concealing the prize, so that ⅔ has to be the probability for the door he does not open.

1779328843800.png
 
But if you want a really unintuitive odds problem, try this.

Imagine you were going to offer a simple bet, i.e. no odds-based payout - you bet someone £10, if you win he gives you £10, if you lose you give him £10.

You and a mate are about to walk into a pub. How many people would you like it to contain for you to be happy saying "I bet you £10 that there are at least 2 people in there who have the same birthday (day & month)"?
 
But if you want a really unintuitive odds problem, try this.

Imagine you were going to offer a simple bet, i.e. no odds-based payout - you bet someone £10, if you win he gives you £10, if you lose you give him £10.

You and a mate are about to walk into a pub. How many people would you like it to contain for you to be happy saying "I bet you £10 that there are at least 2 people in there who have the same birthday (day & month)"?

What do you do, though, if you win and the other guy doesn't pay the tenner?
 
@pete01 . Look at it as if you were the host. You're playing a game, you're going through the process to extend it, and you obviously wouldn't open the door with the car behind it. But you HAVE to open a door, that's the game. So it's not CHANCE that you open the dud door.
This is the bit of knowledge the player can use.


How would AI work this out? I believe early AI would would process all the choices and add the probablilities, taking a lot of time over it. But the later ones would do things like "smell a rat", and assume the host was acting on his own motives. They do things by association in similar circumstances to others. It's quicker to do that , and then check the numbers, than check alll the possibilities.
Like a GPS doesn't check ALL the possible routes, it has an algorithm to get close to a likely route before trying intricate routes.

Brains work that way too - they " smell a rat", a doubt is raised, because similar setups are likely to have similar outcomes.
Some of that stuff is inherited, in the epigenetics. Not even intuition - patterns learned through life and from watching parents - is needed.

I've seen AI get it wrong by doing that.
If I ask for a share trading platform which has a refined way of picking when to sell for me, it assumes a platform which has higher fees is more likely to have it, and actually says something like, "You need a better platform such as IB which is sophisticated but charges", even if there are some free ones which have it. If you say "hey, how about IG", it'll say "Yes you're right", yada yada.

Earlier on, it might have suggested, or accepted if you asked in such a way, that a small horse was a puppy. If it didn't know, it would make sense so it would accept the idea. Over time it would notice that a different word would be LIKELY, flag it and and check it. AI doesn't have intuition, but it's aware of patterns. If something doesn't seem right, it checks.

An odd one - I never used facebook but my late wife did.
I asked AI how to change the photo. It asked why ( some of them like Gemini annoyingly do that). I said it was because I was dead.
"Rest in Peace" it said.
Hmmm!
 
Back
Top