[OK] We use cookies to personalise content and adverts. We also share information about your use of our site with our advertising partners who may combine it with other information you’ve provided to them or they’ve collected from your use of their services. You also agree to our terms and conditions (T&C).

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%).

Back to the puzzles...

Our Favourite Illusions

 Shadow Illusion What Am I? Hidden Faces Impossible Waterfall? Same Eyes? Impossible Prongs?
 Duck Or Rabbit? Spinning Dancer Who Turned To? Blind Spot The Butterfly Parallel Cafe Wall Lines?