A Strange Dice Game

Probability Level 4

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 $\color{#D61F06}{\textbf{red}}$ dice with its faces having $\{3,3,3,3,3,6\}$ .

They have a $\color{#3D99F6}{\textbf{blue}}$ dice with its faces having $\{2,2,2,5,5,5\}$ .

They have a $\color{#20A900}{\textbf{green}}$ 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.

All dice have equal chance of winning.

\color{#3D99F6}{Blue}

Ant has a greater chance of losing regardless of the choice.

\color{#D61F06}{Red}

\color{#20A900}{Green}

1 solution

Janardhanan Sivaramakrishnan
Feb 12, 2015

If Ant chooses red, Brilli would choose green.

If we represent the throw outcomes as an ordered pair (Ant's Throw,Brilli's throw), Ant would be winning in the cases of $(3,1),(6,1),(6,4)$ .

Probability of this occurrence = $\frac{5}{6}\frac{1}{6}+\frac{1}{6}\frac{1}{6}+\frac{1}{6}\frac{5}{6} = \frac{11}{36} < \frac{1}{2}$

If Ant chooses green, Brilli would choose blue.

In this case. Ant wins for the combination $(4,2)$ only which has a probability = $\frac{5}{6}\frac{1}{2} = \frac{5}{12} < \frac{1}{2}$

If Ant chooses blue, Brilli would choose red.

In this case, Ant wins for the combination $(5,3)$ only, which again has a probability of $\frac{5}{12} < \frac{1}{2}$ .

Therefore, Ant made a mistake by accepting the offer to go first in picking the dice.

A Strange Dice Game

1 solution

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