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Two friends are playing with dice. Hector rolls two 20-sided dice and Peter rolls one 20-sided dice.

If Peter's number is between the Hector's two numbers, then Peter wins. Otherwise, Hector wins.

What is the probability of Peter winning?

[Ref: ZYKI]

If any two of the three numbers are the same then Hector wins - as Peter's number has to be between Hector's.

It doesn't actually matter in which order Hector and Peter roll. If we look at the three rolls randomly, then there is a 1÷3 chance that the winning number is Peter's.

It's easiest to see if Peter's rolls first. There are 19 numbers that the Hector's first dice could be. Once Hector's first dice has been rolled there are 18 numbers that his second dice could be. Therefore the probability that the three numbers are different is:

19 x 18
-------
20 x 20

So the overall chance of Peter winning is:

1     19 x 18      57
-  x  -------   =  --
3     20 x 20     200

If the dice were actually 6-sided, then the calculation would be 1÷3 x 5x4÷6x6 = 5÷27 (18.5%).

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