Variations on the Sleeping Beauty

This post won’t di­rectly ad­dress the Sleep­ing Beauty prob­lem so you may want to read the above link to un­der­stand what the sleep­ing beauty prob­lem is first.

Half*-Sleep­ing Beauty Problem

The as­ter­isk is be­cause it is only very similar to half of the sleep­ing beauty prob­lem, not ex­actly half.

A coin is flipped. If it is heads, you are wo­ken up with 50% chance and in­ter­ro­gated about the prob­a­bil­ity of the coin hav­ing come up heads. The other 50% of the time you are kil­led. If it is tails you are wo­ken up and similarly in­ter­ro­gated. Given that you are be­ing in­ter­ro­gated, what is the prob­a­bil­ity that the coin came up heads? And have you re­ceived any new in­for­ma­tion?

Dou­ble-Half*-Sleep­ing Beauty problem

A coin is flipped. If it is heads, a coin is flipped again. If this sec­ond coin is heads you are wo­ken up and in­ter­ro­gated on Mon­day, if it is tails you are wo­ken up and in­ter­ro­gated on Tues­day. If it is tails, then you are wo­ken up on Mon­day and Tues­day and in­ter­ro­gated both days (hav­ing no mem­ory of your pre­vi­ous in­ter­ro­ga­tion). If you are be­ing in­ter­ro­gated, what is the chance the coin came up heads? And have you re­ceived any new in­for­ma­tion?

Dou­ble-Half*-Sleep­ing Beauty prob­lem with Known Day Variation

EDIT: This prob­lem should have said: As above, but when­ever you are be­ing in­ter­ro­gated you are told the day. You may wish to con­sider this prob­lem be­fore the above one.

Sleep­ing Cou­ples Problem

A man and his iden­ti­cal-val­ued wife have lived to­gether for so many years that they have reached Au­mann agree­ment on all of their be­liefs, in­clud­ing core premises, so that they always make the same de­ci­sion in ev­ery situ­a­tion.

A coin is flipped. If it is heads, one of the cou­ple is ran­domly wo­ken up and in­ter­ro­gated about the prob­a­bil­ity of the coin hav­ing come up heads. The other is kil­led. If it is tales, both are wo­ken up sep­a­rately and similarly in­ter­ro­gated. If you are be­ing in­ter­ro­gated, what is the prob­a­bil­ity that the coin came up heads? And have you re­ceived any new in­for­ma­tion?

Sleep­ing Clones Problem

A coin is flipped. If it is heads, you are wo­ken up and in­ter­ro­gated about the prob­a­bil­ity of the coin hav­ing come up heads. If it is tails, then you are cloned and both copies are in­ter­ro­gated sep­a­rately with­out know­ing whether they are the clone or not. If you are be­ing in­ter­ro­gated, what is the prob­a­bil­ity that the coin came up heads? And have you re­ceived any new in­for­ma­tion?

My ex­pec­ta­tion is that the Dou­ble-Half Sleep­ing Beauty and Sleep­ing Clones will be con­tro­ver­sial, but I am op­ti­mistic that there will be a con­sen­sus on the other three.

Solu­tions (or at least what I be­lieve to be the solu­tions) will be forth­com­ing soon.