Deceptive Dice

The Terrible Twins, Innumeratus and Mathophila, were bored.
‘I know,’ said Mathophila brightly. ‘Let’s play dice!’
‘Don’t like dice.’
‘Ah, but these are special dice,’ said Mathophila, digging them out of an old chocolate box. One was red, one yellow and one blue.
Innumeratus picked up the red die. ‘There’s something funny about this one,’ he said. ‘It’s got two 3’s, two 4’s and two 8’s.’
‘They’re all like that,’ said Mathophila carelessly. ‘The yellow one has two 1’s, two 5’s and two 9’s – and the blue one has two 2’s, two 6’s and two 7’s.’
‘They look rigged to me,’ said Innumeratus, deeply suspicious.
‘No, they’re perfectly fair. Each face has an equal chance of
turning up.’
‘How do we play, anyway?’
‘We each choose a different one. We roll them simultaneously, and the highest number wins. We can play for pocket money.’ Innumeratus looked sceptical, so his sister quickly added: ‘Just to be fair, I’ll let you choose first! Then you can choose the best die!’
‘Weeelll . . . ’ said Innumeratus, hesitating.

Should he play? If not, why not?

(Source: Professor Stewart's Cabinet of Mathematical Curiosities, Ian Stewart)

This is one of those instances where averages don't really mean anything. Every one of the dice has the exact same expectation: $\frac{1}{3}(3 + 4 + 8) = \frac{1}{3}(1 + 5 + 9) = \frac{1}{3}(2 + 6 + 7) = \frac{1}{3}(15) = 5$. On this account, the game seems totally fair. So, should Innumeratus play? The names of the twins are a big hint: he should not. The way to know this is by pitting the dice against each other. For any two dice, the set of equiprobable outcomes of the game is the Cartesian product of the distinct numbers on the faces of the dice. For instance, the outcomes of casting the red die vs. the yellow die are $\{3, 4, 8\} \times \{1, 5, 9\} = \{(3, 1), (3, 5), (3, 9), (4, 1), (4, 5), (4, 9), (8, 1), (8, 5), (8, 9)\}$. In five out of these nine cases, the value of the yellow die exceeds that of the red die. Mutatis mutandis, the blue die beats the yellow die and the red die the blue die, five times out of nine. The situation described above is somewhat like playing rock-paper-scissors after you already know the opponent's choice, although more slyly, because the outcome is probabilistic rather than dead certain and, at that, nearly fair. Mathophila was attempting to lead her twin brother on with this offer to pick the "best" die, because the best die is the one chosen knowing what the other player has.

Another somewhat similar gimmick in probability that is apparently often successful in siphoning money out of people is the standard game of craps.