Odds of set over set are???

[HoldenFoldem]

I have had this happen 4 times this week and can hardly believe it. I thought it was an unlikely scenario. Any idea what the actual odds are?

[infinity235]

http://www.math.sfu.ca/~alspach/mag86/

1 to 166 for a four-player game, not changing much when more players join the table.

[Cobra]

I think we sometimes underestimate the probability of a set over set. What we really need to think about is how wide is the range of hands that our opponent played and how does the flop fit into that range. An example might help.

At a full ring table the UTG raises to 3x the BB. He is straight forward TAG and you put his range at AA-TT, AJ's+, AQ'o+, KQ's. This is around 70 combination of hands. The button calls. He is a player that is reasonably tight and understands position you put him on the same range as the UTG plus all pairs, except AA and KK, and some small suited connectors. His range will be about 130 combos. You look down in the big blind and see 22. You and all your opponents have >200 BB so you decide to close out the action and call for set value.

The Flop comes T 6 2, rainbow. You have your set and now you are trying to figure out it anyone has a higher set.

The UTG could only have a set of Tens, the Button could have a set of Tens or sixes. How often did this occur. Because of card removal both opponents ranges have changed slightly after the flop. We will say that UTG has a new range of about 66 combinations and the button has a new range of 120 combinations.

Probability that UTG or button has you beat is,

3/66 + 3/120 + 3/120 = 9.5% or 1 in 10.5 times.

This is meant to just be an estimate, what I am really trying to say is that the wider a persons range is the less likely he is to have a set. If a person has a very tight range and you have bottom set, and both other cards fit his range then there is a 6 in (whatever his range is) chance you are beat.

Cobra

[sickofants]

For argument's sake, let's say that you have middle set (say the flop is J72 and you have a set of sevens) and are against one opponent who's range you can miraculously, with 100% accuracy, narrow down to all pairs, KJ+ and AJ+.

Ie. there are 134 combinations of cards that he could hold under this range (unless I have made some stupid counting error). For set over set to occur, your opponent has to have a set of 2s (you win) or a set of Js (he wins). There are 3C2=3 combinations of cards that could yield each possible set. So if you have middle set, then the probability of set over set occuring is 6/134 or about 4.5%. The probabilities that you lose set over set and win set over set are each 2.25%.

Having typed all that I've just read Cobra's very similar response and feel a little stupid for doing all that (but I've spent too much time to just not post it [img]/images/graemlins/tongue.gif[/img]). I perhaps ought to read all responses before posting myself.

As a guide:

(a) I don't think you should ever lay down any set on a rainbow board against one opponent.

(b) I don't think you should lay down bottom set on a rainbow board against multiple opponents without a great read.

(c) I don't think you should ever lay down middle set because bottom set is as likely a possibility (probably not strictly true but close enough), which you crush.

(d) You should only lay down top set when playing against me.

[tvta]

i go with school of thought that says i dont fold any set (given no made draws), if i go broke then i go broke. about 99.999% of your opponents couldnt make that laydown and it really dosent affect your winrate in my opinion. its such a tough decision that its just better to gamble and go with it. plus hitting a long shot 1 outer is fun [img]/images/graemlins/smile.gif[/img]

so my calculations go like this

to be dealt a pair - 6% or 17-1
flop a set when dealt a pair - 12%

so (1/17)*0.12 = 0.007
or ~140-1

so i just square this 140-1 to get some really small probability of 19600-1

so about every 20k hands dealt assuming every pocket pair sees the flop. notice this is independent of table size (6 handed, full ring...etc). if you want the frequency of how often this is dealt at a table per say of each shuffle and deal, just divide by players getting cards.

i think i did that right. any thing i did wrong?

[HoldenFoldem]

infiniti, Cobra, and Sickof

Thanky you very much for the detailed replies. Infiniti's reply appears to be the most mathematical and yields an answer of 1:167 My week with it happening 4 times must be extra ordinalily extraoridnary. may it never happen again.

Mean while, when i get a set i am going to the felt. I should be right 166 or 167 times AGAINST ANOTHER SET. Straights, flushes and full houses of course reduce the winning probablilty of a set, but we are all aware of that already.

Good luck out there.

[Headblader]

I've had it happen a few times, twice in one 3 hour session before. It's annoying but doesn't change my strategy with sets. It's like getting dealt the same hole cards 2 or 3 times in a row, it probably shouldn't happen alot but it's happened to me.

[binions]

[ QUOTE ]
http://www.math.sfu.ca/~alspach/mag86/

1 to 166 for a four-player game, not changing much when more players join the table.

[/ QUOTE ]

That's not what the article says. It says in a 10-handed game, where only 2 players have been dealt pocket pairs preflop, its 1 in 1035 to be set over set on the flop.

It says in 10-handed games where 2-4 players have been dealt distinct pocket pairs preflop, it is 1 in 166 to have set over set on the flop.

Of course, the condition to have multiple pocket pairs dealt preflop is not always present.

Bottom line, if you flop a set, there is far less than a 1% chance someone else has too.

[karlwig]

[ QUOTE ]
i go with school of thought that says i dont fold any set (given no made draws), if i go broke then i go broke. about 99.999% of your opponents couldnt make that laydown and it really dosent affect your winrate in my opinion. its such a tough decision that its just better to gamble and go with it. plus hitting a long shot 1 outer is fun [img]/images/graemlins/smile.gif[/img]

so my calculations go like this

to be dealt a pair - 6% or 17-1
flop a set when dealt a pair - 12%

so (1/17)*0.12 = 0.007
or ~140-1

so i just square this 140-1 to get some really small probability of 19600-1

so about every 20k hands dealt assuming every pocket pair sees the flop. notice this is independent of table size (6 handed, full ring...etc). if you want the frequency of how often this is dealt at a table per say of each shuffle and deal, just divide by players getting cards.

i think i did that right. any thing i did wrong?

[/ QUOTE ]

not if everyone plays every hand. but they don't. and when atight player is seeing the flop, the odds of him having a pair is of course better than 17-1.

but i think the math would be right if you don't count for those practical difficulties... it seems very low though, this has happened to me online more than 3 times, and i have played under 5K hands...

IT'S RIGGED!!omg

[tvta]

hahaha yeah it does seem more frequent than 20k hands. but yes that is assuming every pocket pair sees the flop (if i did it right).