You are asked to play a classic shell game with two shells.
Each round, a ball is placed under one shell, the shells are rapidly, continuously rearranged for 15 seconds with distraction*, and then you are asked to pick which shell contains the ball.
*By "with distraction," all I mean is that the guy shuffling the shells tries to distract you, and sometimes he is successful, so at the end of each round, you tend to have some knowledge of where the ball is, but you are rarely 100% confident.
If you are correct, you get 4 points.
If you are incorrect, you get -1 points.
If you aren't sure which shell has the ball, you are allowed to opt out of the choice, and receive 2 points.
In order to perform the task optimally, how confident do you need to be each round (in the form of a percentage) to guess (i.e., to forgo the automatic 2 points and shoot for 4)?
Please show your work.
(Yeah, this is simple as hell, but my answer is different from a friend's, and I want to be sure which of us is right.)