Benutzer:Gerhardvalentin/m2

Supposition

Eighteen different halls and eighteen different games. Eighteen guests as contestants, and eighteen different hosts.
After all the guests have selected one door, Hosts have opened another door showing a goat and now the hosts are offering to switch.
Question: Will switching be of advantage for the guests?

Every guest, without knowledge about the location of the car, alredy has had selected one of three doors.
In every game there were two goats, but only one car. So the unselected pair of two doors inevitably always did contain at least one goat.
Each host, having known the location of their car, has opened one of the two unselected doors, showing a goat. This event occurs every time, not only if the guest has chosen the car.
Each host, if he had two goats to choose from, not to show further infotmation on the actual location of the car, never was opening one special door above average nor whenever he could. Offer: After having opened one door, each Host now offers the option to switch to the second unchosen and still closed door. Will it be of advantage for the guests to switch?

Car was behind door No.	Guest selected door No.	The unselected pair of two doors contained always at least one goat but in 2/3 also the car	Loss by switching only in 1/3 (only in 6 of 18 cases) only if, by chance = 1/3, the original coice was the only car	Switching will win in 2/3 in 12 of 18 cases: Always if one of the two doors containing a goat was selected	Host has opened one door showing a goat. If he had got two goats: in 50 % doorX and in 50% doorY	Possible effect of switching (will it be of advantage to switch?)
1	1	Goat ⇔ Goat	Goat, no matter whether Host opens door 2 or door 3		2  (not 3)	Car door 1 was selected, switching will hurt.
1	2	Goat ⇔ Car		Car	3	Goat door 2 was selected, switching to door 1 will win the car.
1	3	Goat ⇔ Car		Car	2	Goat door 3 was selected, switching to door 1 will win the car.
2	1	Goat ⇔ Car		Car	3	Goat door 1 was selected, switching to door 2 will win the car.
2	2	Goat ⇔ Goat	Goat, no matter whether Host opens door 1 or door 3		1 (not 3)	Car door 2 was selected, switching will hurt.
2	3	Goat ⇔ Car		Car	1	Goat door 3 was selected, switching to door 2 will win the car.
3	1	Goat ⇔ Car		Car	2	Goat door 1 was selected, switching to door 3 will win the car.
3	2	Goat ⇔ Car		Car	1	Goat door 2 was selected, switching to door 3 will win the car.
3	3	Goat ⇔ Goat	Goat, no matter whether Host opens door 1 or door 2		1 (not 2)	Car door 3 was selected, switching will hurt.
1	1	Goat ⇔ Goat	Goat, no matter whether Host opens door 2 or door 3		3 (not 2)	Car door 1 was selected, switching will hurt.
1	2	Goat ⇔ Car		Car	3	Goat door 2 was selected, switching to door 1 will win the car.
1	3	Goat ⇔ Car		Car	2	Goat door 3 was selected, switching to door 1 will win the car.
2	1	Goat ⇔ Car		Car	3	Goat door 1 was selected, switching to door 2 will win the car.
2	2	Goat ⇔ Goat	Goat, no matter whether Host opens door 1 or door 3		3 (not 1)	Car door 2 was selected, switching will hurt.
2	3	Goat ⇔ Car		Car	1	Goat door 3 was selected, switching to door 2 will win the car.
3	1	Goat ⇔ Car		Car	2	Goat door 1 was selected, switching to door 3 will win the car.
3	2	Goat ⇔ Car		Car	1	Goat door 2 was selected, switching to door 3 will win the car.
3	3	Goat ⇔ Goat	Goat, no matter whether Host opens door 1 or door 2		2 (not 1)	Car door 3 was selected, switching will hurt.

Question of Rick

If, not knowing about the location of the car, you pick one of these three doors, for example let's say No. 1, and then the host, who knows what's behind the doors, opens another door, for example let's 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?

Then you will answer:

Quite easy, just let's have a look on the table above.
When having picked one door, then my risk to have picked a goat will be 2/3, and my chance to have picked the car 1/3 only.

The unselected pair of two doors however will hide at least 1 goat, but it will hide the car in considerable 2/3. So the probability to win by switching will rise from 1/3 to 2/3.

And now to the exact door numbers you asked for in your question ("picked No. 1, opened No. 3, offered No. 2"):

Let's again have a look on the table above to show us the answer (no host's telltale bias, see supposition).

See hall 10 (1/18 of all possible constellations): By picking the winning door No. 1, the guest has picked the car, and both unselected doors No. 2 and No. 3 containing goats.
The guest, having selected the winning door No. 1, in only 1/18 the host having opened door No. 3 to show the goat, offering door No. 2, containing a goat also, to switch on.

That's in 1/18 of all possible constellations that the guest will be going to loose the car.

And then see halls 4 and 13 (2/18 of all possible consellations): By picking door No. 1 both guests have picked a goat, the car being behind the offered door No. 2, so the host couldn't open door No. 2 but was forced to open door No. 3, and both guests will win with double chance of 2/18 of all possible constellations.

And yes, it's going to be of advantage for two of three (2/3) to switch their choice from door 1 to door 2, because their chances will double from 1/18 to 2/18 of all possible constellations. Hope they will be going to switch, but who knows. Quite a simple solution. And as to your question ("picked No. 1, opened No. 3, offered No. 2"): The probability to win by switching will be going to double and rise from 1/3 to 2/3, you see.