===== Answer =====

2/3

Number the black cards 1-3. The black cards numbered 1 and 2 are in pile together, and card number 3 is in the pile with the red card. You know that your pile has one black card on top, so the possible combinations are

|  B<sub>1</sub>  |  B<sub>2</sub>  |  B<sub>3</sub>  | 
|  B<sub>2</sub>  |  B<sub>1</sub>  |  R  | 

With three total outcomes, two of them have a black card on bottom. The probability of the bottom card being black is 2/3.