I know how to solve this problem using Baye's theorem but I think it can be solved without using it, please have a look if it is correct.
We may reframe the problem by asking that a deck of 52 cards is placed if first 13 of them are red what is the probability that last card is black. Now we have to chose one card as the last card from 26 black and 13 red and so the probability of last card being black is $\frac{2}{3}$.