First to 10!

Five red cards numbered \(1,2,3,4,5\) and two black cards both numbered 5 are randomly ordered face-down into a pile. The cards are flipped over one at a time until either the sum of the numbers on the red cards is at least 10, or the sum of the numbers on the black cards is at least 10. The probability that the sum of the black cards reaches 10 first can be expressed as \(\frac{a}{b}\) where \(a\) and \(b\) are coprime positive integers. What is the value of \(a + b\)?

Details and assumptions

When the cards are flipped over, you get to see their face value (and hence can calculate the sum).

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