You have Kc Kd, and $1300 in front of you. You raise to $50, and your opponent in the small blind, who also had $1300 when the hand began, re-raises to $300. You believe she has either QQ, AK, or AA. The flop comes Jc Ts 3d, and your opponent goes all-in for $1000. Based on her all-in bet, you now believe that she has either QQ or AA, and suppose that you believe that she is equally likely to have either of these two pairs.
a) If she has QQ, what is your probability of winning the hand?
b) If she has AA, what is your probability of winning the hand?
c) Should you call? (Let X = the number of chips you will have after the hand. Calculate the expected value of X if you call, and compare with the expected value of X if you fold.)
I’ve been trying to solve this problem but with no luck. So far I’ve listed
(K,9),(8,9),(Q,2-10),(Q,Q) as cards that your opponent could use to win. Now I have no idea how to proceed from here.

4) You have Kc Kd, and $1300 in front of you. You raise to $50, and your opponent in the small blind, who also had $1300 when the hand began, re-raises to $300. You believe she has either QQ, AK, or AA. The flop comes Jc Ts 3d, and your opponent goes all-in for $1000. Based on her all-in bet, you now believe that she has either QQ or AA, and suppose that you believe that she is equally likely to have either of these two pairs.
a) If she has QQ, what is your probability of winning the hand?
Winning Hands (for opponent)
Hands (K,9),(8,9),(Q,2),(Q,4-8),(QQ),(Q,J),(A,K),(Q,3),(Q,10)
No. of Hands 2*4 4*4 2*4 2*4*5 1 4*3 4*2 2*3 2*3
Total = 99. Therefore P(Win)=1-(99/(45 choose 2))=1-(99/990)=.9
b) If she has AA, what is your probability of winning the hand?
Winning Hands (for you)
Hands (K,2),(K,3),(K,4-9),(K,10),(K,J),(A,Q),(Q,9),(K,K)
No. of Hands 2*4 2*3 2*6*4 2*3 2*3 2*4 4*4 1
Total = 99. Therefore P(Win)= 99/(45 choose 2)=.1
c) Should you call? (Let X = the number of chips you will have after the hand. Calculate the expected value of X if you call, and compare with the expected value of X if you fold.)
Expected Value =X*.9-X*.1+X*.1-X*.9=0
You have nothing to lose, so you should call


Sorry for the messed up formatting


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Since I'm guessing you're interested in the answers to these questions because you're a poker player, and not because you're interested in the mathematics, you should check out PokerStove: Poker Software and Analysis or StoxEV. The answers to a lot of the more complicated Poker questions are really cumbersome matters from an analytic mathematics standpoint, and you're better off using software.

I just glanced over some of the questions, but assuming in part (c) that you did everything right, calling would be (i) maybe okay in a cash game or in a tournament where you are far off from the final table, but I tend to pass in break even situations against inferiors, and (ii) a HUGE blunder in a sit-and-go or near the end of a tournament.

I'm really not much of a poker player. I was just getting into probability and this was one o the questions that came my way.