Poker Articles

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

Introduction

I'm still here. Due to overwhelming demand, I have returned to bore and amaze you with maths and mayhem.

By the way, why do Americans call it math? Is it because you do less of it than us? Let me know.

Today we shall look at a very interesting subject. But first, the answer to my last article's little conundrum. If you haven't figured it out yet, do that now.

Before I give you the answers, here's a disclaimer. Please read it then send me back your signed copy by post before reading further. Reading the rest of this article without reading or returning this disclaimer implies acceptance of the terms.

Disclaimer

I thought I better add this, as I suspect a lot of you are Americans and a few of you are Mathematicians. Worse still, a few of you may be American Mathematicians.

All the calculations in this article are rounded down and may in very rare cases have logical errors. I hereby revoke all liability for injuries caused emotionally, physically and financially to readers basing life and poker decisions on this information.

Answers to the Last Conundrum

Question 1: With which hold 'em hand are you most likely to flop a full house: 5-4 or 4-4?

4-4 is the answer. It's obvious, actually. 4-5 can only make two types of house!

Question 2: Given the answer to Question 1, roughly how much more likely are the chances of flopping the full house? (e.g. three times, four times, etc., etc.)

The answer is about 10.67 to 1. Did you get that right? Were you surprised?

Flopping Top Set: How 'Nutty' Is It?

THIS IS REALLY INTERESTING! Get Ready...

I was playing in a £100 dealer's choice game at Luton the other day. The selected game was six-card Omaha Hi-Lo. Okay, okay, don't dwell on 'Mad Dogs and Englishman.' We all admit that we're crazy on this island. I want to talk to you about something interesting!

The blinds were £2 and £2 on the button, and I limped two to his left holding AKKxxx. The dealer was the wildest player at Luton, and I knew he would raise on the button, which he did. The pot was three-way including me.

The flop came king high and rainbow; I led with £25 and got both callers. The fourth card was a blank apart from giving two clubs. I bet £75. The wild man made it £200 to go. The high blind passed and I re-raised all-in, adding another £125 to the pot.

The button called immediately and neither of us improved on the river, even though that card made a three flush on the board.

I scooped the whole pot.

I tell you the whole story just so you can bask in my glory, but the question that I want to consider is this:

Q1) If you flop top set and the nuts, what is the chance of your hand remaining the nuts on the river without improving?

Analysis

As usual, we'll do the analysis for hold 'em!

N1) Remember, in all the statements below, we assume Q1 is true.

This is quite interesting. I have found out some interesting stuff, like :-

P1) If you hold 22-JJ there is NO chance of retaining the nuts on the river! (See N1)

(Perhaps another reason why QQ, KK, AA are the only premium pairs?)

You are wondering how I got here!

Well I started by finding all five card combinations with no straight and no pair. You might like to pause and think about that.

You will see below how I gradually built the list. Here it is:

A 2 6 7 J
A 2 6 7 Q
A 2 6 7 K
A 2 6 8 J
A 2 6 8 Q
A 2 6 8 K
A 2 6 9 J
A 2 6 9 Q
A 2 6 9 K
A 2 7 8 Q
A 2 7 8 K
A 2 7 9 Q
A 2 7 9 K
A 2 8 9 K
A 3 7 8 Q
A 3 7 8 K
A 3 7 9 Q
A 3 7 9 K
A 3 8 9 K
A 4 8 9 K
2 3 7 8 Q
2 3 7 8 K
2 3 7 9 Q
2 3 7 9 K
2 3 7 T Q
2 3 7 T K
2 3 7 J Q
2 3 7 J K
2 3 8 9 K
2 3 8 T K
2 3 8 J K
2 3 8 Q K
2 5 7 T Q
2 5 7 T K
2 6 7 J Q
2 6 8 J K
2 7 8 Q K
2 7 9 Q K
3 4 8 9 K
3 4 8 T K
3 4 8 J K
3 4 8 Q K
3 6 8 J K
3 7 8 Q K
3 7 9 Q K

Just as an exercise try doing this starting with a 4!

Do you see the high card in each case? Q, K or Ace. This explains P1

All of this gives us the following:-

P2a) If you hold AA the probability that you retain the nuts will be at most 0.48%.

P2b) If you hold KK the probability that you retain the nuts will be at most 0.46%.

P2c) If you hold QQ the probability that you retain the nuts will be at most 0.14%.

The reason for the 'at most' statement is that we are not considering the possibility that three or more cards are suited. For example if we hold Ac-As and the board finishes Ad-2h-6h-7h-Js (as in the list), we do NOT have the nuts. There's a flush on board (but no straight).

To quantify this consideration here's an interesting point:

P3) Given any five chosen cards, the chance of drawing three or more of one suit is 37.11%.

So roughly speaking, you can adjust the figures in P2 downwards by about that amount (roughly one third).

P4) Holding AA, if a '5' or 'T' hits the board, the proposition has no chance.

Just try and find one set of five cards with an A5 or AT and no straight!

Conclusion

Flopping top set (even when it's the nuts) in big draw games such as low-limit hold 'em (as in the US) or 6-card wild games (as in the UK) is NOT such a big hand as you might think!

By the way, when that 3rd flush card fell on the river, I had to use the tried and trusted remedy of standing up to ensure my wild friend hadn't hit a backdoor flush.

Finally, Another Conundrum

What's the probability of flopping a rainbow?

Thanks for your attention.