Nick Szeremeta: Why a flush beats quads!
Thursday, 29 September 2011
The ranking of hands in poker – that is, what beats what – is based on the frequency of the hand-type in a deck of cards. The less frequently a particular category of hand occurs – ie the fewer ways it can be produced – the harder it is to obtain.
It therefore follows that a hand which is difficult to make beats one which is easier to make. For example, one pair occurs more frequently than four of a kind. To suggest that a pair should “beat” four of a kind at poker is clearly nonsense.
However, the number of cards in the deck can alter the mathematics of hands occurring. In strip-deck poker (with the twos, threes, fours, fives and sixes removed) there are 32 cards used for play and a low straight is A-7-8-9-10.
The rank of hands using a 32-card deck is different to that in a full 52 card deck and this has been acknowledged by poker rooms and online card rooms – but only partially.
In fact, everyone has got it wrong for the last 60 or 70 years.
The current hand rankings for poker using the 32-card deck (both online and in bricks and mortar casinos) is as follows:
1) Straight flush
2) Four of a kind
4) Full house
It can be seen that the operators have promoted a flush from ranking lower than a full house to higher, compared with the rankings using a full 52-card deck.
However, according to the principles upon which hand rankings are based, a flush should beat four-of-a-kind. The reason is that a flush occurs less frequently (can be made in fewer ways) than four-of-a-kind in a 32 card deck.
Here’s the mathematical proof:
The number of ways in which four-of-a-kind can be made is as follows:
Four cards of the same rank (sevens through to aces – eight ranks in total), with any one of the remaining 28 cards = 8 x 28 = 224. This means that there are 224 possible ways to make four-of-a-kind from a 32-card deck.
The number of ways in which a flush can be made is as follows:
There are eight available cards in each suit. The number of ways in which a single suit can make a flush is calculated is:
8 x 7 x 6 x 5 x 4 divided by 5 x 4 x 3 x 2 x 1. That is, 6,720, divided by 120 = 56.
As there are four suits, the total number of ways in which a flush can be made with the 32 card deck is therefore 56 x 4 = 224.
It does not take a mathematical genius to notice that this is exactly the same number as there are sets of four of a kind, so it might be assumed that, at the very least, a flush and four-of-a-kind should tie.
Except for one thing…
In poker, a straight flush is classified as a category of its own. Of the possible 224 flushes, 20 of them (five in each suit to the ten, jack, queen, king and ace) are straight flushes.
Therefore, there are only 204 flushes (using the definition of five random cards of the same suit) which can be made from the 32-card deck.
This is not advanced maths; it is an easy sum. As it’s more difficult to make a flush than to make four-of-a-kind (around 10% more difficult), it therefore follows that the holder of a flush against quads should have the pot pushed in his or her direction. (As an aside, it is reasonable to think that big pots will be generated when two big hands like these clash).
As meerkat Aleksandr Orlov would say, “Simples”.
But it isn’t.
Vast sums of money will have changed hands and gone to the player who should have lost the pot if the principles of poker had been adhered to.
Even more money is going to be pushed in the wrong direction unless both the Internet card rooms and the brick and mortar casinos who offer ace to 7 games change their hand rankings.
Nic Szeremeta has been playing poker since the early sixties. On January 1st, 1999 (the same day the Euro currency came into existence), he launched Poker Europa magazine and later that year created the format for Late Night Poker for Channel 4, co-commenting on the first three series.