XY vs. XY

I was sitting today in my mathematics course and trying to figure out a simple probability problem. AKs vs 22, this is not very complicated but for some reason I am drawing a blank. Can some one out there help me out?

pocket pair is ahead about 52 to 48 against overcards before the flop.

I think he's more interested in how to come to that conclusion.

-'rilla

let's play this hot/cold where it's heads up, AK vs 22, and both players know what the other has

there are 6 cards in the deck which improve AK:

So you have 6 chances on each draw of the card out of the deck for an A or K to be picked, thus you end up with:

(6/48)(6/47)(6/46)(6/45)(6/44)

Now, there are 2 cards which improve 22, so on each draw there is are 2 chances for a 2 to come out:

(2/48)(2/47)(2/46)(2/45)(2/44)

There are some other long-shot ways for AK to win:

Board includes QJT,
board has 2 pair, both of which are higher rank than 2
board has full house with cards higher rank than 2
board has 4 of a kind
board has a 4 flush with the suit of either the A or K

All of these are remote possibilities.

This should get you started. The rest is just calculus.

It's actually quite compliicated. That's why you usually use a simulator.

If you aren't going to simulate it - start by throwing out all the "push" boards, like 5 spades (neither of you have the suit) or 6789T. This will throw off your final numbers a bit, but whatever. Lets also skip the straight flushes, somewhat unlikely, especially since neither of you have suited connectors

Total number of 5-card boards = 48*47*46*45*44 = 205476480
So, on a blank board, 22 wins, calculate the number of boards that improve no one.

So, A or K and no 2 boards (AKo wins with pair):
OK, so first card is A or K (6) and remaining cards are not AK2 (40*39*38*37) = 3036960. BUT this can happen 5 ways (A/K can be in any position on board), so you have to multiply this by 5. = 15184800.

Note that this INCLUDES such boards as AJQT9 (broadway straight) and
Awith4 suited to 22, and A3456. First one doesn't matter (still counted as a win for AKo), but we should subtract the others.

Actually, we better start from the best hands and work our way down...

Quads, won by 22:
22XYZ: 2*1*46*45*44 = 182160, 5 choose 2 possibilities, 5!/3!*2! = 10,
total 1821600.

Quads won by AK:
AAAXY, KKKXY = 3*2*1*45*44 (*2) = 23760, 10 possibilities 237600

Full Houses, won by 22:
Possible boards: XXXYZ, 2XXYZ - where X is not A/K/2

So, X can be 3,4,5...Q 10 possible. so 10*(4*3*2*45*44) = 475200
and 10 ways to combine them, 4752000

2nd possiblilty 10 (for all possible XXs) * 2*4*3*45*44 = same, same number of combos 475200

Just keep on going down, Boats won by AK, Flushes, Straights, trips,etc.

Divide each category by 205476480, sum up the Probabilities for each side, and you should get ~52:48.

Proof of this is left as an exercise to the reader.

I think they way you approach this one is to calculate the odds of the AK improving and subtract the odds that the 22 improves. Both the AK and the 22 have the same likelihood of hitting the flush so that cancels out. The AK will improve to a pair 62.5% by the river.
(6/50)+(6/49)+(6/48)+(6/47)+(6/46)= .625543

The 22 will make trips 20.8% of the time if it goes to the river.
(2/50)+(2/49)+(2/48)+(2/47)+(2/46)=.208514

If you subtract chances of catching trips on the 2's from the chances of spiking a pair on the AK you get 41.7 %. Excluding a straight, your AK will win 41.7% of the time. That means 58.3%, (100- 41.7), of the time the twos will hold up (Excluding a Straight). The AK will make a straight slightly more often than the 2's. I would have trouble doing that math as you would have to calculate the odds of every possible scenario of getting that straight. Like flopping three to the straight and catching a runner runner. Or flopping four to the straight and catching the fifth card on the turn or river ect. Maybe someone here can show a simpler way to run the numbers on the straights. Then of course if your AK is suited, you have a slight advantage of making that flush as well. This should get you started in the right direction.

Use a sim.

thanks for the replys, i had an idea it was somewhere around there i was just trying to get an accurate counth, thanks everyone!!