Poker Articles

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

1. Introduction

I have been placed in my little local rebuy comp in Luton five times in succession. Hasn't done me masses of good as there are a lot of 7th and 8th places therein. One Gbp1000+ split first was good though.

The last comp was a novelty 3-card Irish in deference to St Patricks day. In this hold'em variation you hold 3 cards before the flop and discard one after betting on the flop. i.e. You bet twice with three cards.

My fleeting thought was, "If you take a 'loose' players perspective how do these multi card games stack up mathematically.

This article is late and concise. Sorry about the former but I have had a major major week (fund raising for new biz) and I just got slowed up. If you are still too bored to read the maths then skip to the conclusion. Its an obvious one but in Europe where we play lots of Dealers choice with 2-cards being rare and 4-6 cards being common it's a lesson worth learning.

2. The Stats

Firstly lets look at the case where the starting hand played is pure rags.

What's the chance you will hit two pair or above on the flop for this case?

The answer is,

Ta da da da da :-

Table 1 - Probability of Hitting a decent flop

Cards	Hands	Two Pair	Trips	F/H	Quads	Total	Nflop	P
5	5 Rags	2880	480	90	5	3455	16215	21.31%
4	4 Rags	1944	432	54	4	2434	17296	14.07%
3	3 Rags	1080	360	27	3	1470	18424	7.98%
2	2 Rags	396	264	9	2	671	19600	3.42%

The loose players love the games with more cards; There chances get higher and higher.

As an example in Omaha the chance of flopping two pair or more with rags is 14.07%. Much higher than hold'ems meagre, 3.42%.

I have even thrown in a 5-card variation just to extend the argument and believe me in Europe we play it. In fact 6 card Omaha happens and in Finland I once saw 3 guys playing 11-card Omaha.

Anyway onwards.

The chance of having a pair in your 'hole' cards is :-

Table 2 - Pair in the hole

Hole Cards	Pair
5	51.91%
4	32.44%
3	16.94%
2	5.88%

Same lesson. Loose players love those multi-card games. In Finland they play a lot of Omaha and the games can be wild. You get quite a few 'loose-gooses' and as we see 32.44% of the time these birds hold a pair.

Of course if we only consider the 'bigger' pairs (say 99-AA) we have :-

Table 3 - Big Pair in the hole

Hole Cards	99-AA
5	19.96%
4	14.97%
3	7.82%
2	2.71%

The numbers aren't quite as huge but we are not allowing for the huge range of other half hands.

What's the chance of holding a suited ace ? Here's the answer :-

Table 4 - Chance of holding a suited ace

Hole Cards	Suited Ace
5	16.88%
4	13.14%
3	8.47%
2	3.62%

Now lets say we see a flop and there is a scare card. Say the flop is QQ2 and we have TT (with an x, xy or nought). We are first to act. What's the pure chance that someone has the 'Q'.

Here's the table :-

Table 5 - Chance opponent has scare card on flop

Hole Cards	1 Opponents	2 Opponents	3 Opponents	4 Opponents	5 Opponents
5	21.67%	40.70%	57.08%	70.82%	81.92%
4	17.17%	32.73%	46.67%	58.99%	69.70%
3	12.75%	24.64%	35.65%	45.80%	55.07%
2	8.42%	16.47%	24.14%	31.45%	38.39%

For example, in Omaha, with four seeing the flop (and in Finland this happens a lot), 46.67% of the time one of your opponents will have the dreaded 'Q'. You can bet but if you are raised what do you do ?

If we extend the scare card concept to the turn we have :-

Table 6 - Chance opponent has scare card on the turn

Hole Cards	1 Opponents	2 Opponents	3 Opponents	4 Opponents	5 Opponents
5	31.00%	54.82%	72.41%	84.72%	92.68%
4	24.88%	45.24%	61.55%	74.25%	83.79%
3	18.70%	34.91%	48.81%	60.58%	70.39%
2	12.49%	23.89%	34.26%	43.64%	52.08%

and by the river :-

Table 7 - Chance opponent has scare card by river

Hole Cards	1 Opponents	2 Opponents	3 Opponents	4 Opponents	5 Opponents
5	39.41%	65.84%	82.50%	92.17%	97.14%
4	32.03%	55.67%	72.54%	84.06%	91.51%
3	24.37%	44.00%	59.53%	71.58%	80.72%
2	16.47%	30.81%	43.22%	53.89%	62.97%

3. Conclusion

Its obvious, isn't it.

In the 2+ hole-card variations of poker be more selective with your starting hands and when you decide to go, play it strong. Its simple.

Loose players play with more than 2-cards because it gives them more action. Only give them that action when you have the goods.

The problem with these action games is that it gets boring being an observer when the bullets fly but the moral here is 'keep your head down'. A bullet in the eye is more interest than you require.