Assuming I understand your problem, I think the fundamental answer is that you're looking at the wrong event. The event is

two heads **given that** the first coin already landed

And the probability for this event is indeed 1/4. But you've selected the event you're asking for based on the result of the first flip, and then didn't condition on the result of the first flip. Because of this, the event is less meaningful than you think.

So again, it's true that this event has probability 1/4. But it is not the case that this implies mail-mi gets a bad deal. We know this because, if we simulated the whole experiment (of you flipping a coin, then asking mail-mi to bet), he would actually win the bet half the time, and this is consistent with the fact that the probability for the event above is 1/4.

There's also a reductio ad absurdum of this principle. Say I shuffle deck of cards and put them all in front of me, face-down. I look at one card; it's an eight of clubs. I ask mail-mi to bet on this card being an eight of clubs, at 50/50 odds. From mail-mi's perspective, given that he doesn't know the cards, this is only a 1/52 chance. But of course, he will win the bet every time, so

card is eight of clubs

is not the event that measures how good the bet is for mail-mi. Rather, the correct event is

card is {whatever I've just seen as I've looked at it}

And similarly in your case, the event that matters is

both coins land on {what ashersky is asking} **given that** the first coin has been flipped but I don't know the result

and that has probability of 1/2.

So I guess, if there's a lesson here, it's something like "if you select an event based on information, but then don't condition on that information, the probability of that event doesn't mean what you think it means".