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Topic: 3 doors, a key, a car and a goat. (Read 4293 times) |
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Grimbal
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3 doors, a key, a car and a goat.
« on: Jan 4th, 2012, 6:08am » |
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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.
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| « Last Edit: Jan 4th, 2012, 9:12am by Grimbal » |
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SMQ
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Re: 3 doors, a key, a car and a goat.
« Reply #1 on: Jan 4th, 2012, 8:40am » |
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It looks to me like the best strategy is just don't both open the same two doors and the couple will take home the car half of the time.
The reason it isn't four times in nine is that the two events: "Player 1 finds the key" and "Player 2 finds the car" are not independent.
--SMQ
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towr
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Re: 3 doors, a key, a car and a goat.
« Reply #2 on: Jan 4th, 2012, 8:56am » |
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My strategy would be 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
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| « Last Edit: Jan 4th, 2012, 9:01am by towr » |
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rmsgrey
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Re: 3 doors, a key, a car and a goat.
« Reply #3 on: Jan 4th, 2012, 8:58am » |
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on Jan 4th, 2012, 8:56am, towr wrote:| My strategy would be 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 |
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You think there's a 2/3 chance that the goat is behind the left-most door?
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alien2
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Re: 3 doors, a key, a car and a goat.
« Reply #5 on: Jan 4th, 2012, 1:41pm » |
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The doors are soundproof, right? A goat often bleats.
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Grimbal
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Re: 3 doors, a key, a car and a goat.
« Reply #6 on: Jan 5th, 2012, 12:26am » |
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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?
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SMQ
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Re: 3 doors, a key, a car and a goat.
« Reply #7 on: Jan 5th, 2012, 5:31am » |
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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: 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! So together they can win the car 2/3 of the time--a clearly-optimal strategy.
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rmsgrey
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Re: 3 doors, a key, a car and a goat.
« Reply #8 on: Jan 5th, 2012, 7:15am » |
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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...
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Grimbal
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Re: 3 doors, a key, a car and a goat.
« Reply #9 on: Jan 6th, 2012, 1:59am » |
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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?
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SMQ
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Re: 3 doors, a key, a car and a goat.
« Reply #10 on: Jan 6th, 2012, 5:23am » |
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Since the players learn nothing from one another, Player 3's strategy is just 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. The three of them together still have a 2/3 chance of winning.
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Grimbal
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Re: 3 doors, a key, a car and a goat.
« Reply #11 on: Jan 9th, 2012, 9:48am » |
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Correct.
Where I wanted to arrive is:
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.
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| « Last Edit: Jan 9th, 2012, 9:54am by Grimbal » |
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cartoonle
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Re: 3 doors, a key, a car and a goat.
« Reply #12 on: Dec 14th, 2012, 1:23am » |
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Does the player 2 know how many doors played 1 opened?
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Grimbal
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Re: 3 doors, a key, a car and a goat.
« Reply #13 on: Dec 14th, 2012, 7:44am » |
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I decide that .... no.
Anyway, it seems that this information doesn't allow player 2 to improve the odds.
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cartoonle
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Re: 3 doors, a key, a car and a goat.
« Reply #14 on: Dec 16th, 2012, 11:48pm » |
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on Dec 14th, 2012, 7:44am, Grimbal wrote:I decide that .... no.
Anyway, it seems that this information doesn't allow player 2 to improve the odds. |
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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.
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