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Let the hats of monkeys A, B and C be a, b and c respectively. 
This table shows the ways in which the monkeys can select hats.

a b c
a c b
b a c
b c a
c a b
c b a

In only two of the six ways do none of the monkeys have the same hat as when they arrived, hence the required probability is


This problem is taken from the UKMT Mathematical Challenges.