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

118/147,

So 265,do tell if it is right

wrong

To find the probability that the sum of the black cards reaches 10 first, we can use a recursive approach.

Let's define P(i, j) as the probability that the sum of the black cards reaches 10 first, given that the current sum of the red cards is i and the current sum of the black cards is j.

We start with P(0, 0) = 0 since we haven't flipped any cards yet, and the sums are both 0.

Now, for each flip, we have two possibilities:

1. If we flip a red card numbered k (k ≤ 5), the new sum of the red cards becomes i' = i + k. We update P(i, j) as follows:
P(i, j) = P(i', j), for all i < 10
P(i, j) = 1, for all i ≥ 10

This means that if the sum of the red cards reaches 10 or more, the probability for the black cards to reach 10 first is 1.

2. If we flip a black card numbered 5, the new sum of the black cards becomes j' = j + 5. We update P(i, j) as follows:
P(i, j) = P(i, j'), for all j < 10
P(i, j) = 0, for all j ≥ 10

This means that if the sum of the black cards reaches 10 or more, the probability for the black cards to reach 10 first is 0.

We can calculate the values of P(i, j) using dynamic programming. Starting from the base cases P(0, 0) = 0 and P(i, j) = 1 for i ≥ 10, we can iteratively calculate P(i, j) using the formulas above.

Using this approach, we can calculate P(i, j) for all possible pairs (i, j) where 0 ≤ i ≤ 15 and 0 ≤ j ≤ 10. The final probability we're looking for is P(0, 10).

To find the value of a+b, we need to calculate the sum a + b, where a and b are coprime positive integers that represent the probability P(0, 10) as a fraction in lowest terms.

I'll now calculate the values of P(i, j) and find the probability P(0, 10).