Benutzer:Gerhardvalentin/m

Effects of switching (all configurations)

Supposition: The Guest, having no knowledge about the location of the car, selects one door.
There is only one car. So the unselected pair of two doors inevitably contains at least one goat.
The Host, knowing the location of the car, opens one of those two unselected doors, showing a goat. This event occurs every time.
The host, if he has two goats to choose from, not to show further infotmation on the actual location of the car, 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 contains 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 coice was the only car	Switching wins in 2/3 in 6 of 9 cases: Always if one of the two doors containing 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 switching:
1	1	Goat ⇔ Goat	Loss, no matter whether Host opens door 2 or door 3		2  or  3	Car door 1 was selected, switching hurts.
1	2	Goat ⇔ Car		Prize	3	Goat door 2 was selected, switching to door 1 will win the prize.
1	3	Goat ⇔ Car		Prize	2	Goat door 3 was selected, switching to door 1 will win the prize.
2	1	Goat ⇔ Car		Prize	3	Goat door 1 was selected, switching to door 2 will win the prize.
2	2	Goat ⇔ Goat	Loss,  no matter whether Host opens door 1 or door 3		1  or  3	Car door 2 was selected, switching hurts.
2	3	Goat ⇔ Car		Prize	1	Goat door 3 was selected, switching to door 2 will win the prize
3	1	Goat ⇔ Car		Prize	2	Goat door 1 was selected, switching to door 3 will win the prize
3	2	Goat ⇔ Car		Prize	1	Goat door 2 was selected, switching to door 3 will win the prize
3	3	Goat ⇔ Goat	Loss, no matter whether Host opens door 1 or door 2		1  or  2	Car door 3 was selected, switching hurts.
Chance to win:	${\tfrac {1}{3}}$	$0$           ${\tfrac {2}{3}}$	Distribution of chances 1/3 : 0 : 2/3 is in effect from the start until the end of the game, according to the rules
Risk of loss:	${\tfrac {2}{3}}$	$1$           ${\tfrac {1}{3}}$	Distribution of risk  2/3 : 1 : 1/3 is in effect from the start until the end of the game, according to the rules