A Strange Dice Game

Ant and Brilli are playing a game of dice with strange looking dice. There are three dice and each will pick one of them. On each thrower of the higher number gets one point. The game ends after 6000 throws.

The three dice have the following construction.

They have a \(\color{Red}{\textbf{red}}\) dice with its faces having \(\{3,3,3,3,3,6\}\).

They have a \(\color{Blue}{\textbf{blue}}\) dice with its faces having \(\{2,2,2,5,5,5\}\).

They have a \(\color{Green}{\textbf{green}}\) dice with its faces having \(\{1,4,4,4,4,4\}\)

Note that all the three dice have an average face value of \(\frac{21}{6}\).

Ant is the first to choose a dice which gives him the best chance of winning, followed by Brilli choosing a dice which maximizes his winning probability.

Which dice should Ant choose.

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