A Study of Suited Connectors
In Hold'em, there are 56 ways that one can be dealt "suited connectors" (SCs) between 45s and TJs. This is just over 2% of all possible starting hands; on average you will be dealt SCs once every 47 hands.
It's no secret that suited connectors can be a very good, playable starting hand. Because of the coordination of the cards, there can be many ways to make a good hand. When you do make a good hand, it is often very well disguised from your opponents. Additionally, SCs are easy hands to get away from when you miss the flop. For this reason, in general, SCs can make a lot of money when you win and lose little when you don't.
So the question is:
How likely are we to have a "successful" flop when holding suited connectors? This thread is an attempt to answer that question.
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We'll look at the likelihood of a) making a strong hand on the flop and b) making a strong draw on the flop. A
strong hand is considered 2 pair or better. A
strong draw is considered an open-ended straight draw (OESD), flush draw (FD), or combination of the two. For this analysis, we will not consider 1 pair a strong hand; we will not consider an inside straight draw (ISD), a backdoor str8 draw or a backdoor flush draw as strong draws.
I'll analyze each type of "successful" flop (i.e. flops that give us a strong hand or strong draw) separately to figure out the % chance of that flop occuring. At the end, I'll summarize with important and useful figures that we get from this study.
I will try to analyze each "successful" flop similarly.
I'll always assume our hole cards are
.
The math does not change for any SCs (45s- TJs).
Here are some terms I'll use:
Combinations: There are 3 cards to a flop. A combination is the number of ways a successful flop can happen.
and
are 2 (of many)
combinations of flops that give us 2 pair.
Orders: There may be several ways that 3 of same combination are in a flop. These are orders.
and
are 2 (of several) different
orders of the same
combination.
Methods: Not sure how to describe this (note: I'm not a stats guru, so my terminology is not orthodox). It's like this:
and
are 2 different methods of making a str8 flush with 78s.
Also, some hands are described as
Bare. This means that there are no additional good draws beyond the description. e.g. A
BARE STRAIGHT is a made str8 with no flush draw.
For some of the more detailed parts of this analysis, I have linked to an addendum to go deeper into the math. Basically this is to make this thread seem more impressive and to make it easier for you to ignore these parts.
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There are 117600 combinations of flops.
I. 2 PAIR
To flop 2 pair (exactly 2 pair... i.e. no trips, quads... etc.) we need:
a) one of 6 cards to pair up
b) one of 3 cards to hit pair #2
c) one of 44 blanks
6 x 3 x 44 = 792 <--
combinations of order #1
6 x 44 x 3 = 792 <--
combinations of order #2
44 x 6 x 3 = 792 <--
combinations of order #3
TOTAL combinations of 2 PAIR = 2376
% probability of flopping 2 PAIR = 2.02%
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II. MONSTERS (Quads, Full House, Trips)
A) QUADS
To flop quads we need:
a) one of 6 cards to pair up
b) one of 2 cards to trip up
c) one of 1 card to make quads
6 x 2 x 1 = 12
TOTAL combinations of QUADS= 12
% probability of flopping QUADS= just over zilch
B) FULL HOUSE
To flop a full house there are 2 methods:
A)
a) one of 6 cards to pair up
b) one of 2 cards to hit trips
c) one of 3 cards to fill up
B)
a) one of 6 cards to pair up
b) one of 3 cards for 2 pair
c) one of 4 cards to fill up
6 x 2 x 3 = 36
6 x 3 x 4 = 72
TOTAL combinations of FH = 108
% probability of flopping FULL HOUSE = 0.09%
C) TRIPS
To flop trips we need:
a) one of 6 cards to pair up
b) one of 2 cards for trips
c) one of 44 blanks
6 x 2 x 44 = 528
6 x 44 x 2 = 528
44 x 6 x 2 = 528
TOTAL combinations of TRIPS= 1584
% probability of flopping TRIPS = 1.35%
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III. FLUSHES
A) STRAIGHT FLUSH
To flop a straight flush we need:
a) one of 3 cards in the SF
b) one of 2 cards in the SF
C) the final card of the SF
3 x 2 x 1 = 6 combinations
Adjustment: multiply 6 by 4 different methods of getting a str8 flush = 24
Addendum
TOTAL combinations of STRAIGHT FLUSH = 24
% probability of flopping STRAIGHT FLUSH = 0.02%
B) FLUSH
To flop a flush we need:
a) one of 11 remaining spades
b) one of 10 remaining spades
c) one of 9 remaining spades
11 x 10 x 9 = 990
Adjustment: subtract out the 24 of these that are SFs = 966
TOTAL combinations of (non-str8) FLUSH = 966
% probability of flopping (non-str8) FLUSH = 0.82%
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IV. STRAIGHTS
To flop a straight we need:
a) one of 12 coordinated cards
b) one of 8 coordinated cards
c) one of 4 coordinated cards
12 x 8 x 4 = 384
Adjustment: multiply 384 by 4 different methods of getting a str8 = 1536; they are the same methods as a straight flush
Addendum
Adjustment: subtract the 24 Straight Flushes
TOTAL combinations STRAIGHT = 1512
% probability of flopping STRAIGHT = 1.29%
Note: there are straights that can be made that have good draws for improvement... i.e. straights with a Flush Draw. We will get into this later.
There are 1512 combinations of flopping a STRAIGHT.
Subtract 108 combinations of flopping an OESFD + STR8.
Subtract are 108 combinations of flopping a FD + STR8.
TOTAL combinations of a BARE STRAIGHT = 1296
% probability of flopping BARE STRAIGHT = 1.10%
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V. FLUSH DRAWS
A) FLUSH DRAW
To flop a flush draw we need:
a) one of 11 remaining spades
b) one of 10 remaining spades
c) one of 39 non-spades
11 x 10 x 39 = 4290
11 x 39 x 10 = 4290
39 x 11 x 10 = 4290
TOTAL combinations FLUSH DRAWS = 12870
% probability of flopping FLUSH DRAW = 10.94%
Note: not all Flush Draws are created equally. Let's break it down.
A) OPEN-ENDED STR8 FLUSH DRAW (OESFD)
To flop an OESFD we need:
a) one of 2 cards of correct suit and rank
b) one of 1 card or correct suit and rank
c) one of 33 cards that a) do not complete flush and b) do not complete straight
2 x 1 x 33 = 66
2 x 33 x 1 = 66
33 x 2 x 1 = 66
Adjustment: multiply 198 combinations by 3 methods of making an OESFD = 594
Adjustment: add the 12 combinations of making a "Double Bellybuster Straight Flush Draw" = 606
Addendum
TOTAL combinations OESFD = 606
% probability of flopping OPEN ENDED STR8 FLUSH DRAW = 0.52%
B) OPEN-ENDED STR8 FLUSH DRAW AND MADE STRAIGHT (OESFD + STR8)
To flop an OESD + STR8 wee need:
a) one of 2 cards of correct suit and rank
b) one of 1 card or correct suit and rank
c) one of 6 non-suit cards to complete the straight
2 x 1 x 6 = 12
2 x 6 x 1 = 12
6 x 2 x 1 = 12
Adjustment: multiply 36 combinations by 3 methods of having OESD + STR8 = 108
Addendum
TOTAL combinations OESFD + STR8 = 108
% probability of flopping OESFD + STR8 = 0.09%
C) (non-SF) FLUSH DRAW AND MADE STR8 (FD + STR8)
To flop a FD + STR8 we need:
a) one of 2 str8 cards of spades
b) one of 1 str8 cards of spades
c) one of 3 non spade cards to complete str8
2 x 1 x 3 = 6
2 x 3 x 1 = 6
3 x 2 x 1 = 6
Adjustment: multiply 18 combinations by 6 different methods of flopping a FD + STR8 = 108
Addendum
TOTAL combinations FD + STR8 = 108
% probability of flopping FD + STR8 = 0.09%
D) FLUSH DRAW + OPEN ENDED STRAIGHT DRAW (FD + OESD)
To flop a FD + OESD (different than an OESFD) we need:
a) one of 6 non-spade OESD cards
b) one of 1 spade str8 cards to complete OESD
c) one of 8 non-str8 spades
6 x 1 x 8 = 48
6 x 8 x 1 = 48
8 x 6 x 1 = 48
Adjustment: Multiply 144 times the 3 different methods of flopping FD + OESD = 432
Addendum
TOTAL combinations FD + OESD = 432
% probability of flopping FD + OESD = 0.37%
E) BARE FLUSH DRAW
There are 12870 flush draws.
Subtract 606 OESFDs.
Subtract 108 OESFD + STR8.
Subtract 108 FD + STR8.
Subtract 432 FD + OESD.
TOTAL combinations of BARE FLUSH DRAWS = 11616
% probability of flopping BARE FLUSH DRAW = 9.88%
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VI. OPEN ENDED STR8 DRAW (OESD)
A) OPEN ENDED STR8 DRAW (OESD)
To flop an OESD we need:
a) one of 8 str8 cards
b) one of 4 str8 cards
c) one of 40 blanks
8 x 4 x 40 = 1280
8 x 40 x 4 = 1280
40 x 8 x 4 = 1280
Adjustment: multiply 3840 combinations by 3 methods of flopping OESD = 11520
Adjustment: add 2304 combinations of a "Bellybuster Str8 Draw" = 13824
TOTAL combinations of OESD = 13824
% probability of flopping OESD = 11.76%
B) BARE OESD
There are a total of 13824 OESD combinations.
Subtract out 606 OESFDs.
Subtract out 432 FD + OESD.
TOTAL combinations of BARE OESD = 12786
% probability of flopping BARE OESD = 10.87%