Magic Dice Quiz Answer

A brother and sister are stuck inside one weekend because the weather is awful. The young boy is getting bored and his elder sister causally produces a set of dice from her toy box and suggests that they play a game. The boy looks at the dice very suspiciously, and with some justification. The faces on the dice are

Blue - 0, 0, 4, 4, 4, 4

Green - 3, 3, 3, 3, 3, 3

Red - 2, 2, 2, 2, 6, 6

Yellow - 1, 1, 1, 5, 5, 5

"They look rigged" he says. "No" she replies, "they are not rigged. Each face on each dice has an equal probability of coming up. Just to prove I'm not cheating, for each game you can choose your die first, so I can't pick the best one."

Should he play?

Well, only if his sister chooses first every other game. Then it would be fair.

These types of dice are known as Non-transitive, and there are several sets around. This particular combination was invented by Bradley Efron. With this set Blue beats Green, Green beats Red, Red beats Yellow and Yellow beats Blue, each with a probability of 2/3.

We can see that with Blue and Green the Green die must land on a 3. So Blue wins with a 4 (probability 2/3) and loses with a 0 (probability 1/3).

Similarly with Green and Red the Green die must still land on a 3. So Green wins if Red comes up 2 (probability 2/3) and loses if Red is a 6 (probability 1/3).

For Red and Yellow, If Red comes up 6 (probability 1/3) Red wins whatever happens with Yellow (probability 1). If Red comes up 2 (probability 2/3) Red wins if Yellow is 1 (probability 1/2). So adding the two cases:

$$ (\frac{1}{3} * 1) + (\frac{2}{3} * \frac {1}{2}) = \frac{2}{3}$$

In a similar way for Yellow and Blue, the probability of Yellow winning is

$$ (\frac{1}{2} * 1) + (\frac{1}{2} * \frac {1}{3}) = \frac{2}{3}$$

If he plays the game as his sister suggests he will lose his pocket money.


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