interesting coin flip problem

Here is a problem from a decision theory class that I am taking:
If you are told that two coins have been flipped, and that one of them landed heads, what is the probability that the other landed tails?

It seems to me that it would be 1/2, because the two events are independent. However, my teacher (a very intelligent man) INSISTS that the probability is 2/3. Here is his explanation: There are four possibile outcomes for flipping two coins:
HH
HT
TH
TT
Each are equally likely to occur before the trials. He says that if we know that one coin has landed heads, we have eliminated TT from the possibility set. This leaves HH,TH, and HT. In two of the three pairs, the other coin would have to be tails, meaning (according to him) P (tails) = 2/3.
I disagree. Once we know that one coin has landed heads, then obviously TT is eliminated, leaving HH,HT, and TH. However, just because these three pairs were equally likely to occur (probability of each = 1/4) BEFORE the trials, does not mean they are equally likely GIVEN THAT WE KNOW ONE COIN HAS LANDED HEADS. That is to say, in the set HH, HT, TH, there are 4 Hs and 2 Ts. 2 of the Hs are found in HH, one is found in HT, and one is found in TH. Therefore, GIVEN that one coin landed heads, the (posterior) probability that it came from HH is 1/2 (because HH contains 1/2 of the Hs). Similarly, the probability that it came from HT is 1/4 and the probability that it came from TH is also 1/4. Therefore there is a 1/2 chance that the other coin landed Heads and a 1/4 + 1/4 = 1/2 chance that it landed tails.
Who is correct?

HE GOT GAME
ZZzzZZzz