Poker Articles

A Bit of Maths: Scare Cards
By Paul Samuel
(The UK's answer to Mike Caro or Lassie)

1. Introduction

Lets say you have raised strongly with KK, get one caller and see an ace high flop. Your question always has to be - do you believe your opponent has the ace.

My contention will be that roughly, 45.65% of the time he does and you should act accordingly. If you limp in the small blind with no action, the chances would be about 19.08% that the big blind has the 'A'. Where does all this nonsense come from I hear you loyal supporters ask; well here goes.

2. Analysis

Let us roughly divide the pre-flop action in three categories:
1 - Heavily raised
2 - Medium raised
3 - slightly raised or no raise

The let us categorise starting hand according to this grading thus:

Table 1- Hand Rankings

nb. We do not take account of completely random hands, that we might meet in the BB or by lunatics because its simply too complex to analyse.

i.e. For the sake of simplicity let us assume that even for a '3' ranked betting pattern pre-flop, you are left with flop viewers who have some sort of hand, as seen in the table.

Lets take some specific cases:

2.1. Hole Cards are KK and flop is AT2

Firstly here is the relative popularity of cards plotted for each ranking:

Figure 1 - Card Spread (Ex 1)
Its interesting how kings do not appear as popular as queens (well at least according to our assumptions).

Now here are the relative probabilities of your opposition holding specific cards.

Table 2 - Card Spread for 1-3 opponents
For example in a heavily raised pot with three opponents the chances are 83.95% that someone holds an ace.

If the pot is slightly raised and you drag in three players the chances are 47.00% that one of the them holds the ace! hmmmm!

Lets try another one.

2.2. Hole Cards are JJ and flop is TT2

We are concerned about the ten. Look at the popularity plot:

Figure 2 - Card Spread (Ex 2)

Tens aren't that hot! See the tables for 1,2,3 opponents against our rankings.

Table 2 - Card Spread for 1-3 opponents
The worst case scenario is 22.22% for a weakly raised pot and three opponents.

Here's another classic example:

2.3. Hole Cards are AK and flop is QT2

Here's the old card spread:

Figure 2 - Card Spread (Ex 3)

and:

Table 2 - Card Spread for 1-3 opponents

2.4. Hole Cards are 88 and flop is 772

We are concerned about the 7. Look at the popularity plot:

Figure 2 - Card Spread (Ex 2)

Sevens are very unpopular! See the tables for 1,2,3 opponents against our rankings.

Table 2 - Card Spread for 1-3 opponents

The worst case scenario is 0.64% - nothing to worry about.

3. Conclusion

We can go on and on adding new example, I've just modelled it with a spreadsheet. The conclusion is directed at beginners.

Do not assume the worst with a slightly scary flop and do not automatically bluff at such a pot out of position. A good player will re-raise you just to put you to the test !