Yes, exactly. You choose either heads or tails. I flip the coin. If it’s tails and matches what you chose, then you win $1 otherwise lose $1. If it’s heads and matches what you chose, you win $2 otherwise you lose $2. Clearly you will choose heads in this case, just like the Sleeping Beauty when betting every time you wake up. But you choose heads because we’ve increased the payout not the probabilities.
If it’s tails and matches what you chose, then you win $1 otherwise lose $1. If it’s heads and matches what you chose, you win $2 otherwise you lose $2.
Both options have the expected value equal to zero though? (0.5⋅1−0.5⋅1 versus 0.5⋅2−0.5⋅2.)
If you choose heads, you either win $2 (ie win $1 twice) or lose $1. If you choose tails then you either win $1 or lose $2. It’s exactly the same as the Sleeping Beauty problem with betting, just you have to precommit to a choice of heads/tail ahead of time. Sorry that this situation is weird to describe and unclear.
Yes, exactly. You choose either heads or tails. I flip the coin. If it’s tails and matches what you chose, then you win $1 otherwise lose $1. If it’s heads and matches what you chose, you win $2 otherwise you lose $2. Clearly you will choose heads in this case, just like the Sleeping Beauty when betting every time you wake up. But you choose heads because we’ve increased the payout not the probabilities.
Both options have the expected value equal to zero though? (0.5⋅1−0.5⋅1 versus 0.5⋅2−0.5⋅2.)
If you choose heads, you either win $2 (ie win $1 twice) or lose $1. If you choose tails then you either win $1 or lose $2. It’s exactly the same as the Sleeping Beauty problem with betting, just you have to precommit to a choice of heads/tail ahead of time. Sorry that this situation is weird to describe and unclear.