Crooked dice?

A pair of crooked 6 sided dice are rolled.

Each die has the following probabilities \(P(n)\) of rolling the number \(n\):

  • \(P(1) = 1/2\)
  • \(P(2) = 1/4\)
  • \(P(3) = 1/8\)
  • \(P(4) = 1/16\)
  • \(P(5) = 1/32\)
  • \(P(6) = 1/32\)

If the probability that they roll the same number is of the form \( \dfrac ab\), where \(a\) and \(b\) are coprime positive integers, find \(a+b\).


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