The Bizzare Dice

A die is selected at random from an urn that contains two six-sided dice. Die number \(1\) has three faces with the number \(2\) while one face each has the numbers \(1, 3,\) and \(4\). Die number \(2\) has three faces with the number \(4\) while one face each has the numbers \(1, 2,\) and \(3\). The first five rolls of the die yielded the numbers \(2, 3, 4, 1,\) and \(4\) in that order. The probability that the selected die was die number \(2\) can be expressed as \(\,\dfrac{p}{q}\). Calculate \(\,p+q\).