Poker Articles

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

1. Introduction

Really been tough dragging this article out this week. So in the tradition of all mathematicians, when in doubt solve something useless.

When I was kid I like a lot of kids used to play sneaky games of poker, mainly 5 card draw or 3 card brag (what a crazy game).

Now on the subject of the former game and reflecting on those distant years (10 years ago at least) what was the maths being draw poker.

Was it better to hold xxy and draw two or to draw to the pair ?

Well without philosophising too much here are the figures I came up with.

2. Analysis

First the probability table for drawing a hand from cold is :-

Table 1 - Hand Probability Chart

All you guys who play Caribbean stud at the casinos note that the 50:1 they pay out on a Royal isn't too generous.

Now consider the cards you might hold before drawing

2.1. Pair

For example you might hold AA and draw three.

Then we have:-

Table 2 - Drawing to a pair

2.2. Trips

For example you hold 666 and draw two.

Then we have:-

Table 3 - Drawing to trips

2.3. pair & one

Perhaps you hold AAK and draw two.

Then we have:-

Table 4 - Drawing to Pair & One

2.4. 3 Flush

Perhaps you hold A48 and draw two.

Then we have:-

Table 5 - Drawing to a 3 flush

2.5. 4 flush

Perhaps you hold A489 and draw two.

Then we have:-

Table 6 - Drawing to a 4 flush

2.6. Open ended Straight

Perhaps you hold 789T and draw one.

Then we have:-

Table 7 - Drawing to an open ended straight

2.7. Gutshot Straight

Perhaps you hold 78TJ and draw one.

Then we have:-

Table 8 - Drawing to a gutshot straight

2.8. AKQ suited

Perhaps you hold AKQand draw two.

Then we have:-

Table 9 - Drawing to AKQ suited

2.9. Two pair

Perhaps you hold AA55 and draw one.

Then we have:-

Table 10 - Drawing to two pair

3. Conclusion

Well here's a couple of deductions from the above:-

If you hold AAKxy, mathematically you are more likely to improve drawing three to the pair rather than holding the AAK and drawing two. In the latter case you are a 4 : 1 to improve but by holding the pair you are a 3 : 1 to improve.

If you hold a low pair and nothing else, you will improve 28.16% of the time but if you draw 5 cards 68.10% of the time you will make some kind of hand. So with 22389 mathematically you should draw five.

It appears that if you hold AKQ99 it is better to draw to the AKQ (improvement 40.8163% of the time) then the 99 (improvement 28.16% of the time).

Well you can dig out your own too; I am sure I will receive mail correcting the numbers and I look forward to that but for now it's goodbye from yours truly.