Suppose an uneven die has the following probability for each side.

p(1) = p(2) = 0.2 + ,

p(3) = p(4) = 0.2 2,

p(5) = p(6) = 0.1 + ,

where is an unknown parameter.

Roll this die 10 times, and the results are '2', '4', '3', '1', '2', '5', '4', '6', '1', '2'. Find the maximum likelihood estimate for . (You'll need to use the quadratic formula.)