These were your exact words:
Her credence when asked is 1/3, because of the four possible cells in the 2x2 array, one is eliminated and only one of the remaining three is Heads. — JeffJo
This can be generalised as:
Her credence when asked is
, because of the
possible cells in the
array,
are eliminated and only
of the remaining
has Outcome X.
In the traditional problem the rules are:
- A coin is tossed
- If the coin lands on tails then Sleeping Beauty is woken on Day 1 and Day 2
- If the coin lands on heads then Sleeping Beauty is woken on Day 1
Given this, and where Outcome X is the coin landing on tails, the values are:
Your reasoning entails that Sleeping Beauty's credence that the coin landed on tails is
.
In a slightly different version of the problem the rules are:
- A 6-sided die is rolled
- If the die rolls a 1 then Sleeping Beauty is woken on Day 1, Day 2, ... and Day 10
- If the die rolls a 2 then Sleeping Beauty is woken on Day 1
- If the die rolls a 3 then Sleeping Beauty is woken on Day 1
- If the die rolls a 4 then Sleeping Beauty is woken on Day 1
- If the die rolls a 5 then Sleeping Beauty is woken on Day 1
- If the die rolls a 6 then Sleeping Beauty is woken on Day 1
Given this, and where Outcome X is the die rolling a 1, the values are:
Given that
, your reasoning entails that Sleeping Beauty's credence that the die rolled a 1 is
.
In my extreme version of the problem the rules are:
- A 2
100-sided die is rolled
- If the die rolls a 1 then Sleeping Beauty is woken on Day 1, Day 2, ... and Day 2
101
- If the die rolls a 2 then Sleeping Beauty is woken on Day 1
- If the die rolls a 3 then Sleeping Beauty is woken on Day 1
...
- If the die rolls a 2
100 then Sleeping Beauty is woken on Day 1
Given this, and where Outcome X is the die rolling a 1, the values are:
Given that
, your reasoning entails that Sleeping Beauty's credence that the die rolled a 1 is approximately
.
Whereas Halfers will say the only thing that Sleeping Beauty needs to consider is the coin toss/die roll, and so her credence in each of these situations is:
- The coin landed on tails:
- The 6-sided die rolled a 1:
- The 2
100-sided die rolled a 1: