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If you have Ah,Kh vs Qc,2s and the flop comes down Qh,Jh,3c, giving your opponent top pair and giving you a flush draw, straight draw and two overcards: How should you play this hand? who is the favorite to win?

Note that while your opponent currently has the best hand, you have as many as 18 outs, most of which will win you the pot. The 18 outs are: nine hearts (Th,9h,8h,7h,6h,5h,4h,3h,2h) three straights (Tc,Td,Ts) three aces (Ac,Ad,As) and three kings (Kc,Kd,Ks). In order for your opponent's queens to hold up, none of these cards can appear among the next two cards. Since there are 45 cards left in the deck, 45-18=27 restricting the last two cards to those 27 cards.

total possible combinations for the last two cards: 45 C 2 = 990
combinations with which your opponent will win: 27 C 2 = 351

Now, there are several cases in which you will catch one of your
eighteen outs and still lose. This will occur if you hit top pair and your opponent hits two pair or trips, or if you hit the flush and your
opponent hits a full house. Note that if you hit the straight, there is no way your opponent can win.

The following 32 combinations contain one of your eighteen outs but you still lose:

Therefore, your opponet will win 351 + 32 = 383 times out of the 990 and you will win 990 - 383 = 607 times. You are thus a 61 percent favorite to win the pot (607/990=.6131).

Now lets assume that the turn card is an Ace or a King. You are now a huge favorite to win since your opponent only has four outs
(Qd,Qs,2d,2s). You will win 91 percent of the time (40/44=.9090).

Now lets assume that the turn card helps neither one of you. You are then a 41 percent underdog to win (18/44=.4090)

Now lets assume that the turn card is a deuce (other then the deuce of heart) giving your opponent two pair. Now you only have eleven outs: eight flush cards (excluding the 2h) and three straight cards. You are a 25 percent underdog (11/44=.25).

Now lets assume that the turn card is a queen giving your opponent
trips (Qh,Jh,3c,Qd). Now you only have ten outs: seven flush cards
(excluding 2h and 3h) and three straight cards. You are a 23 percent underdog to win (10/44=.227).

Now lets assume that the turn card is the 2 of heart giving your
opponent two pair and giving you the flush. Your opponent now has four outs (Qd,Qs,2d,2s). You will win 91 percent of the time (40/44=.9090).

What would you rather see on the turn, the deuce of heart or an
ace/king? Theoretically, you are the exact same mathematical favorite.

However, in practice most players feel vulnerable with just a pair of aces but they fee pretty cocky with a nut flush. Therefore, if you were the one holding the Q2 against what you know is the AhKh, you should be more inclined to call if the deuce of heart hits than if the A/K hits. That's because you can expect to get reraised on the end if you make a full house but your opponent won't reraise your two pair or trips with just aces. There's therefore more implied odds in the Q2.