# Benutzer:Gerhardvalentin/m

## Effects of switching, all possible configurations

### Supposition

The Guest, having no knowledge of the location of the Car, selects one of three doors.

There are two Goats, but there is only one Car. So the non-selected pair of two doors inevitably hides at least one Goat.

The Host, knowing the location of the Car, will always open one of those two non-selected doors to show you a Goat.
This event occurs every time, not only if the Guest has selected the Car.

The Host, if he has two Goats to choose from, will not be opening one special door "above average" nor "whenever he can".

Car
behind
door
Guest
selects
door
The unselected
pair of two doors
hides always
at least one Goat

but in 2/3
also the Car
Loss by switching only in 1/3
(only in 3 of 9 cases):
Only if, by chance = 1/3,
the original choice was the only Car

Switching
wins in 2/3

in 6 of 9 cases:
Always if one
of the two doors
hiding
a Goat was
selected
Host opens a
door showing
a Goat.
If he has got
two Goats:
in 50 % door
X
and
in 50% door
Y
Effect of swapping to the second still closed Host's door
that he always must offer as an alternative:
1 1 GoatGoat Loss, no matter whether Host opens door 2 or door 3 2  or  3  Car door 1 was selected, switching hurts.
1 2 GoatCar   Prize 3  Goat door 2 was selected, switching to door 1 will win the prize.
1 3 GoatCar   Prize 2  Goat door 3 was selected, switching to door 1 will win the prize.
2 1 GoatCar   Prize 3  Goat door 1 was selected, switching to door 2 will win the prize.
2 2 GoatGoat Loss,  no matter whether Host opens door 1 or door 3 1  or  3  Car door 2 was selected, switching hurts.
2 3 GoatCar   Prize 1  Goat door 3 was selected, switching to door 2 will win the prize
3 1 GoatCar   Prize 2  Goat door 1 was selected, switching to door 3 will win the prize
3 2 GoatCar   Prize 1  Goat door 2 was selected, switching to door 3 will win the prize
3 3 GoatGoat Loss, no matter whether Host opens door 1 or door 2 1  or  2  Car door 3 was selected, switching hurts.
Chance
to win:
${\displaystyle {\tfrac {1}{3}}}$ ${\displaystyle 0}$           ${\displaystyle {\tfrac {2}{3}}}$ Distribution of chances 1/3 : 0 : 2/3 is in effect from the
start until the end of the game, according to the rules
Chance to win the Car by swapping to the other door offered is not 1/2 : 1/2.
By staying I will win the Car only in 1/3, if – by chance – I selected the Car.
Risk
of loss:
${\displaystyle {\tfrac {2}{3}}}$ ${\displaystyle 1}$           ${\displaystyle {\tfrac {1}{3}}}$ Distribution of risk  2/3 : 1 : 1/3 is in effect from the
start until the end of the game, according to the rules
Swapping to the other door offered doubles my chance on the Car from 1/3 to 2/3,
it will give me the Car for sure if – in 2/3 – I have selected one of the two Goats.
Regardless of what door the Guest selected and what door was opened by the Host.
The standard scenario of the paradox forever firmly excludes any closer information.

Someone asks you the following question:

If, not knowing about the location of the Car, you pick one of these three doors, for example let's say #1, and then the Host, who knows what's behind the doors, opens another door, for example let's say #3, which has a Goat. He then says to you, "Do you want to pick door #2?" –  Is it to your advantage to switch your choice?