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A Dicey Paradox

Age 11 to 14 Challenge Level:


Four fair dice are marked on their six faces, using the mathematical constants $e$, $\pi$ and $\phi$ as follows:

4 4 4 4 0 0  
B: $\pi \pi \pi \pi \pi \pi$ where $\pi$ is approximately 3.142
C: e e e e 7 7 where e is approximately 2.718
D: 5 5 5 $\phi \phi \phi$ where $\phi $ is approximately 1.618

The game is that we each have one die, we throw the dice once and the highest number wins. I invite you to choose first ANY one of the dice. Then I can always choose another one so that I will have a better chance of winning than you. You may think this is unfair and decide you want to play with the die I chose. In that case I can always chose another one so that I still have a better chance of winning than you. Investigate the probabilities and explain the choices I make in all possible cases.

Does it make any difference if the dice are marked with 3 instead of $\pi$, 2 instead of $e$ and 1 instead of $\phi$?