9. Alberto and Barbara are playing with a die. After a while, they notice that the die is loaded, and the number 1 comes up more frequently than the other 5 numbers (which remain equally probable). They decide that, when a 1 is rolled, that roll is canceled and they roll again. If they continue to roll the die until they get 2 valid rolls, what is the probability that the sum of the 2 valid numbers rolled is 8?
(A) $\frac{3}{25}$
(B) $\frac{1}{6}$
(C) $\frac{1}{5}$
(D) $\frac{6}{25}$
(E) $\frac{1}{4}$.
9. The answer is $(\mathbf{C})$. Since if a 1 is rolled, the die is rolled again, it is as if the die has only 5 faces. Now, to get 8, any initial roll is fine, but on the second valid roll, only the complement to 8 is valid. Therefore, the required probability is equal to that of getting a specific number on the first roll, which is $\frac{1}{5}$.

## Second Solution

The possible ordered pairs of valid results are 25. Of these, only $(2,6),(3,5),(4,4),(5,3),(6,2)$ are valid, so the probability is $\frac{5}{25}=\frac{1}{5}$.