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        riddles >> medium >> 3 doors, a key, a car and a goat.
        
(Message started by: Grimbal on Jan 4^th, 2012, 6:08am)

Title: 3 doors, a key, a car and a goat.
Post by Grimbal on Jan 4^th, 2012, 6:08am

Here is a problem I found on another forum:

You take part in a game. There are 3 doors. Behind one door is a car key, behind another is the matching car, and behind the remaining one is a goat.

Your goal is to win the car. Unlike in the Monty Hall problem, there are 2 items to discover and 2 players.

The first player must discover the key. He selects a door and opens it (unlike the classical problem where the host opens another door for you). If the player doesn't find the key, he can choose another door and open it.

If player 1 discovers the key, player 2 can start.

Player 2 must discover the car. He selects a first door and opens it. If necessary he can select a second door and open it. Player 2 doesn't know what player 1 has seen, not even which door(s) he opened.

If player 1 finds the key and player 2 finds the car, they win the car (assume they are a couple). In all other cases, they loose.

What is the best strategy, and what is the probability of winning? Both players can agree on a strategy before starting to play.

Title: Re: 3 doors, a key, a car and a goat.
Post by SMQ on Jan 4^th, 2012, 8:40am

It looks to me like the best strategy is [hide]just don't both open the same two doors[/hide] and the couple will take home the car [hide]half of the time[/hide].

The reason it isn't [hide]four times in nine[/hide] is that the two events: "Player 1 finds the key" and "Player 2 finds the car" are not independent.

--SMQ

Title: Re: 3 doors, a key, a car and a goat.
Post by towr on Jan 4^th, 2012, 8:56am

~~My strategy would be [hide]to pick one door we both don't open (say left-most); if that's the one with the goat, we win the car. So that gives a win of 2 out of 3 [/hide]~~

Title: Re: 3 doors, a key, a car and a goat.
Post by rmsgrey on Jan 4^th, 2012, 8:58am

on 01/04/12 at 08:56:39, towr wrote:

My strategy would be [hide]to pick one door we both don't open (say left-most); if that's the one with the goat, we win the car. So that gives a win of 2 out of 3 [/hide]

You think there's a 2/3 chance that the goat is behind the left-most door?

Title: Re: 3 doors, a key, a car and a goat.
Post by towr on Jan 4^th, 2012, 9:00am

Well, not anymore.. :-[

Title: Re: 3 doors, a key, a car and a goat.
Post by alien2 on Jan 4^th, 2012, 1:41pm

The doors are soundproof, right? A goat often bleats.

Title: Re: 3 doors, a key, a car and a goat.
Post by Grimbal on Jan 5^th, 2012, 12:26am

It is a cheeky goat. It likes to imitate the sound of a car.

OK, so far SMQ leads with 1/2, and towr follows with 1/3. Can you do better?

Title: Re: 3 doors, a key, a car and a goat.
Post by SMQ on Jan 5^th, 2012, 5:31am

Aha, the players can choose a different second door based on what they find behind the first. WLoG, let's say player 1 opens "door A" first, and "door B" next if they find the car first. Player 1, then, has only two strategies: open door B if they find the goat as well, or instead open door C if they find the goat.

Player 2 has twelve strategies: they can open any of the three doors first, and any of the remaining two next upon finding either the goat or the key.

Of the 24 possible strategies, one is special: [hide]Player 1 opens door A first, door B next if they find the car behind door A, or door C next if they find the goat behind door A; Player 2 opens door B first, door A next if they find the key behind door B, or door C next if they find the goat behind door B. In all cases where Player 1 finds the key, player 2 finds the car[/hide]! So together they can win the car 2/3 of the time--a clearly-optimal strategy.

--SMQ

Title: Re: 3 doors, a key, a car and a goat.
Post by rmsgrey on Jan 5^th, 2012, 7:15am

I had SMQ's strategy, but, for some reason, kept miscounting the cases where it wins - I worked it out as 1/2 rather than 2/3 for some reason...

Title: Re: 3 doors, a key, a car and a goat.
Post by Grimbal on Jan 6^th, 2012, 1:59am

Congratulations!

Now, let's imagine there is a third player who is required to discover the goat, or nobody wins. How well can they do?

Title: Re: 3 doors, a key, a car and a goat.
Post by SMQ on Jan 6^th, 2012, 5:23am

Since the players learn nothing from one another, Player 3's strategy is just [hide]Player 2's strategy with the goat and the car swapped: Open door C first, door A next if they find the key behind door C, or door B next if they find the car behind door C[/hide]. The three of them together still have a 2/3 chance of winning.

--SMQ

Title: Re: 3 doors, a key, a car and a goat.
Post by Grimbal on Jan 9^th, 2012, 9:48am

Correct.

Where I wanted to arrive is:
[hide]If you number the key, the car and the goat as 1, 2, 3, so that player n needs to find item n, then the solution can be expressed as:
Player n opens door n. If he discovers item k<>n, then he opens door k.
It works for the cases where item n is behind door n but also for cases where 2 items were swapped. In these cases, when a player finds an item other than the one he wants, he knows the two have been swaped.
[/hide]

Title: Re: 3 doors, a key, a car and a goat.
Post by cartoonle on Dec 14^th, 2012, 1:23am

Does the player 2 know how many doors played 1 opened?

Title: Re: 3 doors, a key, a car and a goat.
Post by Grimbal on Dec 14^th, 2012, 7:44am

I decide that .... no.

Anyway, it seems that this information doesn't allow player 2 to improve the odds.

Title: Re: 3 doors, a key, a car and a goat.
Post by cartoonle on Dec 16^th, 2012, 11:48pm

on 12/14/12 at 07:44:09, Grimbal wrote:

I decide that .... no.

Anyway, it seems that this information doesn't allow player 2 to improve the odds.

Yes, the car can be won if and only if the first player found the key. So the success can't be more that 2/3 of times. If the second player would know how many gates first player opened, the solution would be a bit easier, but the success rate would still be 2/3.