Poker Articles

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

Introduction

I've decided to write a short article about poker and maths. I could write a long article but I suspect I might lose a few of you before I complete this sentence.

I am a mathematician, computer consultant and a poker player. Poker players will often say to me that maths is far less important than intuition in poker but I say, "All good poker players seem to have an almost instinctive understanding of the expectation involved in any poker situation. I believe that this intuition is simply a cellular reflection of the mathematics involved. I believe that if you surround yourself in the mathematics of poker, you can accelerate your intuition!"

Here's another quote: "The language of humans is speech. The Language of the universe is Mathematics." Good, huh? Ok, enough philosophy, here's lesson number one!

Your intuition can deceive you

I was playing in a £20 rebuy in Luton England a number of months ago when a player picked an argument with me over the chance of completing his flush after flopping a four-flush. "50/50!" he cried in glee after I got all my chips in with his larger stack on the flop. I had top pair and he still had a four-flush. I WON!

I know he was thinking that roughly a quarter of the remaining decks was his suit, and since ¼ + ¼ = ½, hey, presto!

Here's a simple counter argument; If you dealt four further cards (instead of two), would the fact that ¼ + ¼ + ¼ + ¼ = 1 mean you're guaranteed to hit your flush? Of course not. That's like saying you must turn a head if you flip a coin twice.

Lets take the coin analogy as an example to illustrate the maths, then extend it to hold 'em. The chance of hitting a head if you flip a coin twice is equal to:

1 - (the chance of flipping two tails on the trot)

or

1 - ( ½ x ½) = 1 - ¼ = —

i.e. if you flip a (fair) coin twice it is 75% likely that you will hit a head!

Have I lost you yet ? Its not that tough, really. Now lets look at flopping a four-flush in hold 'em. There are nine outstanding cards of our suit and the chance of hitting your flush will be:

1 - (the chance of missing your flush)

And, the chance of missing is the chance of not hitting a suit on the turn and the river, or...

(52 - 5 - 9)   (52 - 6 - 9)
----------- x -----------
  (52 - 5)        (52 - 6)

This may look tricky, but all we've done is say the chance of missing your flush is the chance of missing it on the turn times the chance of missing it on the river. As an illustration, the chance of missing it on the turn is the number of non-flush cards remaining divided by the number of cards remaining!

The number of non-flush cards remaining is 52.

52 - (# of known cards) - (# of flush cards undealt)
= 52 - 5 - 9

The number of cards we could choose from is 52.

52 - (# of known cards, i.e. flop + your hand)
= 52 - 5

The whole equation gives pretty much 65.03%, so we subtract that from 1 and deduce the following: the chance of making your flush after flopping a 4 flush is about 34.97%!

So, Sorry to Mr. X at Luton. Keeping taking that draw though!

Here's another one, but before you look at the answer, answer this question: If you hold J-J, what is the chance of flopping an overcard and no set?

If you don't know the answer take a wild guess. If you do know the answer, don't tell the first guy! The answer is 51.82%.

That's why J-J is not such hot stuff, as a lot of people think! Half the time you get an 'iffy' flop. For those of you interested, here's the same calculation done for all starting pairs.

Probability of At Least one o/c and No set
K-K	20.67%
Q-Q	37.84%
J-J	51.82%
T-T	62.94%
9-9	71.53%
8-8	77.92%
7-7	82.43%
6-6	85.39%
5-5	87.12%
4-4	87.96%
3-3	88.22%
2-2	88.24%

Suited Cards

Heres another famous Luton Quote: "But they were suited."

I'm just throwing this one in, but the the chance of you flopping a flush in hold 'em is about 0.84% or 118 to 1.

The chance of hitting your flush by the river, given that you hold two suited cards is about 6.40% or approximately 15 to 1.

Dump those suited 2-7 hands, guys. Just dump them!

And Finally

I hope I didn't lose too many of you and that this short article is of some interest. In case you want to hear from me again, here's a final conundrum (like all good soaps):

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

Question 2: Given the answer to Question 1, roughly how much more likely are the chances of flopping the house? (e.g. 3 times, 4 times, etc., etc.)

Thanks for your attention.

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