by Konstantin Othmer

This section takes the ideas from Analyzing Betting Behavior in the last issue and combines them with some basic statistics as an example of card reading. The situation is a common one and is as follows: Suppose you are on fourth street and have just made the fourth card to a flush. The pot is small and your lone opponent paired his doorcard. None of your flush suit is out, and your opponent bets. Should you call?

You are the favorite and could even raise if your opponent has a pair, but you are an underdog and should consider folding if he has two pair or trips. But how do you figure this out?

To start the analysis, let’s determine the end points of the range of probabilities. If the player you are up against is on the extreme end of being tight and will only start with a pair or better (assume this player doesn’t play flush or straight draws), then the probability he now holds two pair or better is 100 percent!

What sort of situation defines the low end of the range? A player who is extremely loose (plays every hand) and the other two cards of his doorcard rank were already seen in other hands has the lowest likelihood of having two pair (he cannot hold trips). In this case the probability the player has two pair is about 6 percent (3/51). Wow! The chances range from 6 percent to 100 percent! The real world is rarely as extreme as these examples, of course. To analyze this situation further, first look at the relative number of hands.

Situation	Number of Hands	Number of Hands w/ Paired Dooryard
3rd St. Hands	22,100	0
4th St. Hands	270,725	15,925
4th St. Hands w/One Pair	82,368	13,728
4th St. Hands/w Two Pair	2,808	936
4th St. Hands w/Trips	2,496	1,248
4th St. Hands w/Quads	13	13
4th St. Hands w/Two Pair or Better	5,317	2,197

The table shows there are 22,100 different third-street hands (none of which have a paired doorcard of course!) and 270,725 different fourth-street hands of which 15,925 have a paired doorcard. If a player plays every hand, the majority of the time (86 percent) the fourth-street hand with a paired doorcard has only one pair. As the following tables show, if an opponent is more selective about his starting hands, the chances he holds two pair or better after pairing his doorcard increase dramatically. The following table shows the total number of fourth-street hands with pairs, trips, or quads as well as the number of hands with the doorcard paired.

Situation	No. of Sidecards Exposed	No. of Hands	No. with Paired Dooryards
Two Pair	0	2,808	936
Two Pair	1	1,404	486
Two Pair	2	468	156
Trips	0	2,496	1,248
Trips	1	624	312
Quads	0	13	13

The table shows that if a player starts with a pair on third and pairs his doorcard, his most likely holding is trips (57 percent) if none of his doorcards is exposed, and two pair (60 percent) if one or more is exposed. Both halves of that statement are non-intuitive! But if you think about it for a while, it makes sense. Trips are more likely than two pair, since the player is more likely to have a split pair than a hole pair by a 2-to-l margin. That three cards give two pair and only two give trips is not enough to overcome the higher probability of the split pair. Pretty cool, huh?

But things change when a sidecard is exposed. Now there is only one card that gives trips, and two that give two pair. The differential (2-to-l) is now larger than previously (3-to-2), and two pair is more likely.

But there is another twist. The stats assume the exposed card was removed before the hand was dealt. Clearly, if you deal the deck all the way to the bottom, you will expose all the sidecards and these stats don’t apply. But fourth street is early enough in the hand (you will have seen about 12 cards) where these stats are fairly accurate. As the hand progresses, the chance of the doorcard coming out increases, and it’s not accurate to work backward. If you see two cards of the paired door rank, you know the player can’t hold trips!

The following tables show the number of hands when starting with a three-flush, a three-straight, or a three-straight-flush.

Situation	No. of Sidecards Exposed	No. of Hands
One Pair Anywhere	0	10,008
One Door Pair	0	1,668
One Door Pair	1	1,112
One Door Pair	2	556

If your third street hand is:

Situation: Three-straight (no straight-flush)

Number of Hands: 480

Then on fourth street you will have one of the following:

Situation	No. of Sidecards Exposed	No. of Hands
One Pair Anywhere	0	4,320
One Door Pair	0	720
One Door Pair	1	480
One Door Pair	2	240

If your third street hand is:

Situation: Three-straight flush

Number of Hands: 32

Then on fourth street you will have one of the following:

Situation	No. of Sidecards Exposed	No. of Hands
One Pair Anywhere	0	288
One Door Pair	0	48
One Door Pair	1	32
One Door Pair	2	16

Putting all this data together, it’s possible to figure out the chances a player has two pair or better if he starts with only a pair, flush, and straight hands. The following probabilities were derived from the previous tables. Notice the large effect of exposed cards.

Plays	No. of Sidecards Exposed	Chances of Two Pairs or Better
Everything	2	5.88%
Everything	0	13.8%
Three-straight, Three flush, Pairs	2	16.1%
Three-straight, Three-flush, Pairs	1	22.4%
Three-straight, Three-flush, Pairs	0	45.1%
Only Pairs	N/A	100%

The table shows the chances the player holds two pair or better after pairing his doorcard for various starting hands. Clearly this is a limited example, but based just on knowing the player’s starting requirements and whether some of his needed cards are exposed, you can get a ballpark estimate of his holdings.

If you correlate this data with the psychological response clues, you should be well on your way to becoming a good card reader.