On a remote island, precisely one third of the natives are liars who always lie, precisely one third are the knights who always speak truthfully and the remaining one third are the knaves who strictly alternate between lying and telling the truth.

The chances of encountering any one of the three types of natives on the road on the island are the same.

If a traveler meets a native on the road each of two successive days, what is the probability that at least one of the two natives is a knave?

 There are 3^2=9 couples for 2 days (k+l+N)*(k+l+N).Withot the knaves there are only 2^2=4.

So 5 pairs must include a knave(N):  NN,Nk,Nl,lN,kN.

Another approach : The prob on the 1st day was 1/3, on the 2nd likewise", avoiding counting NN twice we get:

1/3+1/3-1/3*1/3=(6-1)/9=5/9

Posted by Ady TZIDON on 2013-01-15 13:25:14