A crooked die and a straight die
A crooked six-sided die and a fair six-sided die are rolled at the same time.
The crooked 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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