Three dice with no best die

Three special six-sided dice are on the table.

• Die A shows $3$ on every face.
• Die B shows $2$ on four faces and $6$ on two faces.
• Die C shows $1$ on two faces and $5$ on four faces.

Two players each roll one die, and the higher number wins. Your friend gets to pick a die first, and then you choose one of the remaining two. Your friend picks die A.

Which die should you take, and what is your probability of winning?