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The Monty Hall Problem ?

  
? ? ?  
Behind one of these three doors is a car. Behind each of the other two doors is a goat.
Choose which door you want: if you choose the one hiding the car you get to keep it. (virtually)
 
Manual Automatic slow
Automatic medium
Automatic fast
repeating times
OK, your choice is marked. I won't show you what's behind your chosen door yet, but I will open a different door
and show you what's there, then ask you to choose whether to stick to your choice or switch doors.
Press when ready:
Now you know what's behind one of the doors, do you want to change to the other closed door?
Choose whether to or ? (press one of these two buttons)
Results
door 1 Chosen times Switched from times Switched to times won times.
door 2 Chosen times Switched from times Switched to times won times.
door 3 Chosen times Switched from times Switched to times won times.
Switched times, winning times. () Stuck times, winning times. ()

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Imagine that the set of Monty Hall's game show Let's Make a Deal has three closed doors. Behind one of these doors is a car; behind the other two are goats. The contestant does not know where the car is, but Monty Hall does.

The contestant picks a door and Monty opens one of the remaining doors, one he knows doesn't hide the car. If the contestant has already chosen the correct door, Monty is equally likely to open either of the two remaining doors.

After Monty has shown a goat behind the door that he opens, the contestant is always given the option to switch doors.What is the probability of winning the car if she stays with her first choice? What if she decides to switch?

Direct quote from The Math Forum Ask Dr. Math: FAQ

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