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 ab where a and b are coprime positive integers. What is the value of a+b?

To solve this problem, let's consider all the possible ways the cards can be flipped over 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.

There are a total of 7 cards in the pile, and each card can be flipped over in two ways - either with the red side facing up or the black side facing up. So, the total number of possible outcomes can be calculated as 2^7 = 128.

Now, let's calculate the number of outcomes where the sum of the black cards reaches 10 first. We need to consider the order in which the cards are flipped over and keep track of the sum.

To simplify the calculation, we can break it down into cases based on the order in which the black cards are flipped over.

Case 1: The two black cards are flipped over consecutively.
Since both black cards are numbered 5, the only way the sum can reach 10 is if both cards are flipped over consecutively as the first two cards. This can happen in only one way.

Case 2: One of the black cards is flipped over first, followed by the other black card.
To calculate the number of outcomes for this case, we need to calculate the number of ways the two black cards can be placed in positions 3 through 7 in the sequence of flips. This can be calculated as (5 choose 2) = 10.

So, the total number of outcomes where the sum of the black cards reaches 10 first is 1 + 10 = 11.

Therefore, the probability that the sum of the black cards reaches 10 first is 11/128.

To express this probability as a fraction in the form ab, where a and b are coprime positive integers, a represents the numerator (11) and b represents the denominator (128).

The sum of a and b is 11 + 128 = 139.

So, the value of a+b is 139.