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The strange world of poker probabilities

Like most casino games, poker is partly a game of opportunity, or more accurately, probabilities, with the least likely events scoring the highest and returning the biggest wins. Whether you are drawing cards or spinning the roulette wheel, the odds of the outcome are always at the front of your mind. So just what are the chances of drawing the different hands at poker, and are they always reflected in the hand value?

What are the odds?

Logically, the lowest value hand overall in poker, the ‘high’ is also the most common. In fact, with odds of 0.995 to one, you are slightly more likely to draw an ‘high card’ hand than you are to draw one of all the other combinations put together. However, there is a strange anomaly in the high-card hands that we will return to later.

The next highest hand is the pair, which is only slightly less likely than a high-card hand. With 2,860 possible single pair hands in a 52 card deck, rising to over a million combinations if you factor in the four different suits, you will draw a pair around 42% of the time, or once in every 2.37 draws. From here, however, the hands get far less common.

The odds of drawing two pairs are 20 to one, or just 4.75%, and three of a kind is less than half as likely at 46 to one or just 2.11%. That’s around the same chance of finding the love of your life on an airplane according to a study by HSBC Bank in the UK.

Once in a lifetime odds

At the other end of the scale, we have the near impossible odds of being dealt the top hand of a royal flush. Although there are 2,598,960 possible five card hands, there are only four ways to draw a royal flush, making the odds an enormous 649,740 to one. That means you will be dealt a royal flush just 0.000154% of the time. To put this in perspective, you would have to play, on average, twenty hands a day for 89 years to be dealt a royal flush, and even then, there are no guarantees, as each new hand still has a one in 649,740 chance, even if you’ve played for 90 years.

You can play until you’re grey and still not be dealt a royal flush

That might sound unlikely to even happen, yet you can make a mind bogglingly unlikely thing happen right in front of you, right now. Take a pack of cards and shuffle it at random and then spread them out. According to RationalWiki, the odds of the cards being in that particular unique order are a head wrecking:

80658175170943878571660636856403766975289505440883277824000000000000 to one.

A strange anomaly

So far so good. You can see how hand values rise in poker as they decrease in likelihood. This is the same across almost all casino games. Take roulette, for example, a game that is made simple and easy to understand by the way you place your bets and the return you get. Black or red pay even money, as there is a 50:50 chance of the ball landing on that colour (ignoring the house take from the green 0 or 00). Single numbers have a 1 in 36 chance of occurring and so pay 35 times your stake and so on. Simple.

Strangely however, in poker, whether online rooms or live,  the least likely outcome isn’t always the winner. While a full house – 693 to one probability, will always beat a flush – 508 to one probability, a seven high, is around 120 times harder to get than an ace high, yet the ace high will always win. Confused? Just think about how many ways you can make an ace high (which can contain any non consecutive combination of cards), compared to how few ways there are to make a seven high (which can only contain five of the six cards numbered seven through to two with one missing). The seven high should be far superior to the ace high, yet it is the lowest ranked hand of them all.

Of course, different odds mean different things to different people. The odds of you dying in the next year are less than 100 to one, while the odds of winning the lottery are enormous. But I still know which odds I’d rather get the better of this weekend!