Pocket PAir V Pocket Pair Odds

can someone clear up overpair odds for me.

I always understood that say a pocket pair of 10s against say a pocket pair of 5s was about 89% to 11% favourite but looking on other sites they suggest the overpair is about 80 / 20 favourite.

Whilst I am no genius at working out the odds I always thought that the chance of the under pair hitting one of their remaining 2 cards was about 4.5% per card, pre flop they have 5 cards to come so 5 x 4.5% equates to 22% chance of them making trips, however I also have the same chance of hitting the 3rd 10 so this should half the odds making them only an 11% chance of winning.

Am I going wrong somewhere or are these other sites wrong?

I understand that possible straights, flushes, split pots can alter the odds a fraction but I am confused.

I just need to know, am I as unlucky as I think I am or are the odds only 4 to 1 in my favour of winning.

Thanks

Yes, an overpair to an underpair is about a 4:1 favorite. If you check AA vs. KK on a Texas Hold'em calculator like this one, it shows 82.36% vs. 17.09% for pairs of the same suits, and 81.06% vs. 18.55% for pairs of different suits (doesn't quite add up to 100% because there's always a small chance they'll tie).

Your mathematical formulation of this situation is flawed. There's a ~22% chance any pocket pair will hit trips by the river. There's a ~78% chance it won't. So when two people both hold pocket pairs, the probability that by the river, one of them will hit trips and the other won't (which is the condition that usually satisfies the lower pair beating out the higher one) is about 0.22*0.78 = 0.1716 or a little over 17%. Then add to this the possibility of the lower pair beating the higher with a flush, the overpair is much closer to a 4:1 than 5:1 favorite over the underpair.

Great info odds are my weak suit☺

That is 100% correct (~81 /~19) work back from that.

thanks for that but I still cant see how the 22% chnce isnt halved to give 11%

I can see what you are saying about the 22/78 but I am sure matamatically it has to be 11%!!!!!

it is not 11% no matter how confused you may be.

So, agreed that my calculation of 22% is flawed by the chances of the lower hand making a straight or a flush but I cannot see why your calculations and the calculation sites dont seem to work in the fact that the higher pair can also improve, surely there is as much of a chance of the higher pair tripping up as there is the lower pair and likewise for the flush or the straight.

The pair of tens rarely improves.

If you hold TT and hit a set of tens, your hand dosent improve.. you had the best hand before and the ten was a blank to your opponent.

The only time the set of tens comes into play is if your opponent firstly hits a set of 5s and then you redraw.

So calculate the chances of your opponent hitting a set of 5s and THEN you redrawing to a set of tens. (yes it makes a difference but its almost negligible)

My calculations did work in the fact that the higher pair could also improve.

I'll try again. For an underpair to outdraw an overpair with trips, the following two conditions must be met: 1) the underpair makes trips by the river AND 2) the overpair fails to make trips by the river.

The underpair will make trips by the river ~22% of the time. The overpair will fail to make trips by the river ~78% of the time.

To calculate the probability of BOTH events happening, you multiply together the probabilities of each individual event happening. So, 0.22*0.78 = 0.1716.

Note that the probability of both the overpair and underpair making trips is less than 0.22*0.22, or <5%.

Just run it through an odds calculator: (note, suits do have some barring on the odds)
http://twodimes.net/h/?z=8288075

pokenum -h th ts - 5c 5s
Holdem Hi: 1712304 enumerated boards
cards win %win lose %lose tie %tie EV
Ts Th 1381851 80.70 321733 18.79 8720 0.51 0.810
5s 5c 321733 18.79 1381851 80.70 8720 0.51 0.190