User:JumpDiscont
For each of the problems here that start with "There are 3 doors" - that is, all but problem 16 -
- The doors are numbered door 1 , door 2 , door 3 .
- and
- A car is put behind a random one of those three doors.
- and
- Goats are put behind the other two of those three doors.
- and
- The host knows where the car is.
.
Start with Problem 0, and if your answer to that is 1/2, then go to problem 16.
If your answer to Problem 0 is 1/2 and your answer to problem 16 is 1/3, then the instructions are how you can perform binary search to find where you think the answer changes, although if you get annoyed at the problems closer to the end seeming to have nothing to do with Monty Hall, then you might do linear search instead, starting from problem 0.
Problem 0
[edit]From
- There are 3 doors. The host opens a door, chosen as follows:
- It can't be door 1. It can't be the door the car is behind. If the car
- is behind door 1, then the choice is 50/50 between the other two doors.
, what is the has-car probability for door 1?
If your answer is 1/2, then go to problem 16.
Problem 1
[edit]From
- There are 3 doors. The host chooses a door as follows,
- It can't be door 1. It can't be the door the car is behind. If the car
- is behind door 1, then the choice is 50/50 between the other two doors.
- , and then opens the door the host chose.
, what is the has-car probability for door 1?
If your answer to this problem is 1/3 or 1/2 , then I imagine that is
different from your answer to problem 0 or problem 2 respectively.
Problem 2
[edit]From
- There are 3 doors.
- The host chooses a door as follows:
- It can't be door 1. It can't be the door the car is behind. If the car
- is behind door 1, then the choice is 50/50 between the other two doors.
- The host opens the door the host chose.
, what is the has-car probability for door 1?
If your answer to this problem is 1/3, then binary search takes you to problem 1.
If your answer to this problem is 1/2, then binary search takes you to problem 3.
Problem 3
[edit]From
- There are 3 doors. The host chooses a door as follows:
- It can't be door 1. It can't be the door the car is behind. If the car
- is behind door 1, then the choice is 50/50 between the other two doors.
, what is the has-car probability for door 1?
If you got here via my binary search instructions and your answer
to this problem is 1/3 or 1/2 , then that is different
from your answer to problem 2 or problem 4 respectively.
Problem 4
[edit]From
- There are 3 doors. The host chooses an integer as follows:
- It must be one of {1,2,3} . It can't be 1. It can't be
- the number of the door the car is behind. If the car is
- behind door 1, then the choice is 50/50 between 2 and 3.
, what is the has-car probability for door 1?
If your answer to this problem is 1/3, then binary search takes you to problem 2.
If your answer to this problem is 1/2, then binary search takes you to problem 6.
Problem 5
[edit]From
- There are 3 doors. The host chooses an integer as follows:
- If the car is behind door 3, then the host chooses 2.
- If the car is behind door 2, then the host chooses 3.
- If the car is behind door 1, then the choice is 50/50 between 2 and 3.
, what is the has-car probability for door 1?
If you got here via my binary search instructions and your answer
to this problem is 1/3 or 1/2 , then that is different
from your answer to problem 4 or problem 6 respectively.
Problem 6
[edit]From
- There are 3 doors. The host chooses an integer as follows:
- If the car is behind door 3, then the host chooses 5.
- If the car is behind door 2, then the host chooses 6.
- If the car is behind door 1, then the choice is 50/50 between 5 and 6.
, what is the has-car probability for door 1?
If your answer to this problem is 1/3, then binary search takes you to problem 5.
If your answer to this problem is 1/2, then binary search takes you to problem 7.
Problem 7
[edit]From
- There are 3 doors. The host chooses an integer as follows:
- If the car is behind door 3, then the host chooses 5.
- If the car is behind door 2, then the host chooses 5.
- If the car is behind door 1, then the choice is 50/50 between 5 and 6.
, what is the has-car probability for door 1?
If you got here via my binary search instructions and your answer
to this problem is 1/3 or 1/2 , then that is different
from your answer to problem 6 or problem 8 respectively.
Problem 8
[edit]From
- There are 3 doors. The host chooses an integer as follows:
- If the car is behind door 3 or the car is
- behind door 2, then the host chooses 5.
- If the car is behind door 1, then the choice is 50/50 between 5 and 6.
, what is the has-car probability for door 1?
If your answer to this problem is 1/3, then binary search takes you to problem 4.
If your answer to this problem is 1/2, then binary search takes you to problem 12.
Problem 9
[edit]From
- There are 3 doors. The host chooses an integer as follows:
- If the car is not behind door 1, then the host chooses 5.
- If the car is behind door 1, then the choice is 50/50 between 5 and 6.
, what is the has-car probability for door 1?
If you got here via my binary search instructions and your answer
to this problem is 1/3 or 1/2 , then that is different
from your answer to problem 8 or problem 10 respectively.
Problem 10
[edit]From
- There are 3 doors, and the host has a coin.
- The host chooses an integer as follows:
- If the car is not behind door 1, then the host chooses 5.
- If the car is behind door 1, then the choice is 50/50 between 5 and 6.
If your answer to this problem is 1/3, then binary search takes you to problem 9.
If your answer to this problem is 1/2, then binary search takes you to problem 11.
Problem 11
[edit]From
- There are 3 doors, and the host has a coin.
- The host places or flips a coin as follows:
- If the car is not behind door 1, then the host places the coin showing heads.
- If the car is behind door 1, then the host flips the coin.
, what is the has-car probability for door 1?
If you got here via my binary search instructions and your answer
to this problem is 1/3 or 1/2 , then that is different
from your answer to problem 10 or problem 12 respectively.
Problem 12
[edit]From
- There are 3 doors, and the host has a coin.
- If the car is not behind door 1, then the host places the coin showing heads.
- If the car is behind door 1, then the host flips the coin.
, what is the has-car probability for door 1?
If your answer to this problem is 1/3, then binary search takes you to problem 10.
If your answer to this problem is 1/2, then binary search takes you to problem 14.
Problem 13
[edit]From
- There are 3 doors, and the host has a coin.
- If the car is not behind door 1, then the host places the coin showing heads.
- If the car is behind door 1, then the host places the coin showing heads.
, what is the has-car probability for door 1?
If you got here via my binary search instructions and your answer
to this problem is 1/3 or 1/2 , then that is different
from your answer to problem 12 or problem 14 respectively.
Problem 14
[edit]From
- There are 3 doors, and the host has a coin.
- The host places the coin showing heads.
, what is the has-car probability for door 1?
If your answer to this problem is 1/3, then binary search takes you to problem 13.
If your answer to this problem is 1/2, then binary search takes you to problem 15.
Problem 15
[edit]From
- There are 3 doors.
, what is the has-car probability for door 1?
If you got here via my binary search instructions and your answer
to this problem is 1/3 or 1/2 , then that is different
from your answer to problem 14 or problem 16 respectively.
Problem 16
[edit]From
- The host rolls a 3-sided die, with sides {1,2,3} .
, what is the probability that the die's result is 1?
If your answer to problem 0 is 1/2
and your answer to this problem is 1/3 ,
, then binary search takes you to problem 8.