• Win a car! (or a goat ...)
    56 replies, posted
[b]THE PREMISE[/B] [quote]Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?[/quote] [URL=http://filesmelt.com/][IMG]http://filesmelt.com/dl/12716054743651.jpg[/IMG][/URL] [B]THE ASSUMPTION[/b] Now, on the face of it, it looks as though there is a 50/50 chance that you win the car whichever door you choose, and switching doesn't matter. [B]THE TRUTH[/B] If you switch doors, there is a 2/3 chance of you winning the car. [I]What? That is impossible![/i] I hear you cry. Well, here is the solution. [quote]The player, having chosen a door, has a 1/3 chance of having the car behind the chosen door and a 2/3 chance that it's behind one of the other doors. As the host opening a door to reveal a goat gives the player no new information about what is behind the door he has chosen, the probability of there being a car remains 1/3. The new information from the host tell us only that there is a 0/3 chance of the car being behind the revealed door. Therefore, a 2/3 chance remains that the car is behind the other unopened door. Switching doors thus wins the car with a probability of 2/3, so the player should switch[/quote] Another way of looking at it is to imagine the possible outcomes assuming that you switch or don't switch. [b]IF YOU SWITCH[/B] 1) You pick the door with the car (1/3). The host opens one of the other doors at random, revealing either goat. You switch, and [b]lose.[/b] 2) You pick the door with the first goat (1/3). The host opens the door with the other goat ([b]not at random[/b]). You switch, and [b]win.[/b] 3) You pick the door with the second goat (1/3). The host opens the door with the other goat (again, [b]not at random[/b]). You switch, and [b]win.[/b] In the situations where you switch, you win the car [b]two thirds of the time[/b]. [b]IF YOU STAY[/b] 1) You pick the door with the car (1/3). The host opens one of the other doors at random, revealing either goat. You stay, and [b]win.[/b] 2) You pick the door with the first goat (1/3). The host opens the door with the other goat ([b]not at random[/b]). You stay, and [b]lose.[/b] 3) You pick the door with the second goat (1/3). The host opens the door with the other goat (again, [b]not at random[/b]). You stay, and [b]lose.[/b] In the situation where you stay, you win the car [b]one third of the time[/b]. Confusing? [b]VISUALISING THE SOLUTION[/b] [quote]It may be easier to appreciate the solution by considering the same problem with 1,000,000 doors instead of just three. In this case there are 999,999 doors with goats behind them and one door with a prize. The player picks a door. The game host then opens 999,998 of the other doors revealing 999,998 goats—imagine the host starting with the first door and going down a line of 1,000,000 doors, opening each one, skipping over only the player's door and one other door. The host then offers the player the chance to switch to the only other unopened door. On average, in 999,999 out of 1,000,000 times the other door will contain the prize, as 999,999 out of 1,000,000 times the player first picked a door with a goat. A rational player should switch. Intuitively speaking, the player should ask how likely is it, that given a million doors, he or she managed to pick the right one. The example can be used to show how the likelihood of success by switching is equal to (1 minus the likelihood of picking correctly the first time) for any given number of doors. It is important to remember, however, that this is based on the assumption that the host knows where the prize is and must not open a door that contains that prize, randomly selecting which other door to leave closed if the contestant manages to select the prize door initially.[/quote] [b]AN EXPLANATION[/B] [quote][B]The critical fact is that the host does not randomly choose a door - he always chooses a door that he knows contains a goat after the contestant has made their choice. This means that the host's choice does not affect the original probability that the car is behind the contestant's door.[/B] When the contestant is asked if the contestant wants to switch, there is still a 1 in 3 chance that the original choice contains a car and a 2 in 3 chance that the original choice contains a goat. But now, the host has removed one of the other doors and the door he removed cannot have the car, so the 2 in 3 chance of the contestant's door containing a goat is the same as a 2 in 3 chance of the remaining door having the car. [B]This is different from a scenario where the host is choosing his door at random and there is a possibility he will reveal the car.[/B] In this instance the revelation of a goat would mean that the chance of the contestant's original choice being the car would go up to 1 in 2. This difference can be demonstrated by contrasting the original problem with a variation that appeared in vos Savant's column in November 2006. In this version, the host forgets which door hides the car. He opens one of the doors at random and is relieved when a goat is revealed. Asked whether the contestant should switch, vos Savant correctly replied, "[B]If the host is clueless, it makes no difference whether you stay or switch. If he knows, switch[/B]"[/quote] [b]STILL NOT CONVINCED?[/B] Test it yourself by using playing cards and a friend. Take 3 cards from a deck, one ace and two kings. [LIST] [*]Have your friend shuffle the three cards and memorise their order, then place them face down on the table. [*]Pick one of the cards that you think is the ace. [*]Your friend then flips over one of the cards that [b]he knows[/b] is a king and [b]not[/b] the one you specified. [*]Flip over the third card, and record whether it was a king or an ace. [*]Repeat as many times as you like. [/LIST] If you compare the number of times you get the ace compared to the number of times you get the king, it will tend to be 2/3. Source, further information, other visualisations and solutions: [url]http://en.wikipedia.org/wiki/Monty_Hall_problem[/url]
I would want the goat, I like goat milk. On topic: This is some deja vu right here, I could have sworn reading about the exact same scenario here on Facepunch once.
i want to have the car so i can milk the car i tried riding a goat once it wasnt that pleasant
I didn't get it until the If You Stay, If You Switch section. Now it makes sense. Holy mindfuck :psyduck:
I saw this on a TV program on mathematics I think.
If I get the goat, can I still keep it?
[QUOTE=Al_Borland;21430331]If I get the goat, can I still keep it?[/QUOTE] Yes.
Remember reading this somewhere. Weird stuff.
lemme guess just read the curious incident of the dog in the night time
What car can I win?
[QUOTE=Adius Shadow;21430170]I would want the goat, I like goat milk. On topic: This is some deja vu right here, I could have sworn reading about the exact same scenario here on Facepunch once.[/QUOTE] What if it's a male goat
[QUOTE=Laserbeams;21430505]What if it's a male goat[/QUOTE] Inject it with oestrogen.
I remember this from 21.
[QUOTE=Laserbeams;21430505]What if it's a male goat[/QUOTE] You can still milk it :smug: I've read about this problem being submitted in a math magazine a long while back. The magazine got a flood of people complaining that a skilled mathematician was wrong and that they were right :v:
[QUOTE=BrickInHead;21430488]lemme guess just read the curious incident of the dog in the night time[/QUOTE] What? no ..
This may make sense in theory, but in real life I don't think it would apply.
It doesn't matter, these gameshows always rip you off it's probably got bad transmission or its inefficient
I'll pick #3.
I remember this from "The Curious case of the dog in the Night time." Best part of the damn book.
[QUOTE=Penguiin;21430927]This may make sense in theory, but in real life I don't think it would apply.[/QUOTE] Read the last part about the cards.
I remember this from a youtube video.
You stole maverick's avatar.
I remember this from probability class :v:
I can't get my head around this :psyduck:
[img]http://www.smh.com.au/ffximage/2004/09/01/terminal_wideweb__430x285.jpg[/img] medicine for goat
I saw this shit on QI, mang.
Are the car and the goat worth the same? I'd rather have a $50 goat then a $50 car.
We did this in maths class, I had heard of it before from "The Curious Incident of the Dog in the Night Time" so I knew the answer already, our maths teacher knew the answer but didn't understand it, so I got to go up to the board to try and explain it :smug:
This is difficult to wrap my head around, but it makes sense if you think about for any length of time.
It's intuitive if you think about how high your chances are for being wrong rather than being right.
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