I love 5! Sorry for being unfair.

Discrete Mathematics Level pending

I have an unfair \(6\)-sided die. Its faces were numbered with integers \(1\), \(2\), \(3\), \(4\), \(5\), and \(6\). I rolled the die \(2\) times. Let \(A\) be the value of the first roll and \(b\) be the value of the second roll. The probability that \(A\) and \(B\) will be both equal to \(5\) \(OR\) \(A\) and \(B\) will be distinct integers is \(\frac{241}{245}\). If the smallest possible probability (minimum value) of rolling a \(5\) in the die is in the form \(\frac{m}{n}\) where \(m\) and \(n\) are coprime positive integers. Find \(m + n\)

Details and Assumptions:

  • The die in this problem is unfair. Each of the probabilities of rolling a \(1\), \(2\), \(3\), \(4\), \(5\), and \(6\) are not all equal to each other.

  • If I get \(2\) in the first roll and \(5\) in the second roll, then \(A = 2\) and \(B = 5\)


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