The chances of different hands of playing cards

Check the chances of POKER hands

If you like BIG numbers then you'll love this!

The Murderous Maths Book "Do You Feel Lucky?" calculates and explains the chances of getting freak hands of playing cards in games such as bridge and poker. In bridge you are dealt 13 cards, in poker you usually only work with five cards. There's more about poker hands here.

If you are dealt 13 cards, your chances of getting the following hands are:

• 13 spades (in any order): 1 in 635,013,559,600
• 13 cards in the same suit (in any order): 1 in 158,753,389,900
• 13 spades in the CORRECT order (i.e. Ace,2,3,....queen,king): 1 in 3,954,242,643,910,000,000,000

If you have four players receiving 13 cards each, these are the chances involved:

• All four players getting one complete suit each: 1 in 2,235,197,406,900,000,000,000,000,000
• All four players getting all 13 cards of their favourite suit in the correct order :
  1 in 80,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

Here are a few more chances that we didn't have room for in the book.

DISTRIBUTION

For a lot of games where you are dealt 13 cards (such as Bridge and Whist) it's handy to know how many cards you're likely to get in the same suit. You might think the most likely distribution is to have 3 suits with 3 cards each and one suit with 4 cards. (This is described as 4-3-3-3) However you might be surprised! Here's the distribution list starting with the most likely....

• 4-4-3-2 : 21%
• 5-3-3-2 : 15.5%
• 5-4-3-1 : 13%
• 5-4-2-2 : 10.6%
• 4-3-3-3 : 10.5%
• 6-3-2-2 : 5.6%
• 6-4-2-1 : 4.8%
• 6-3-3-1 : 3.4%
• 5-5-2-1 : 3.2% (as shown in the picture)
• 4-4-4-1 : 3%
• 7-3-2-1 : 1.9%
• 6-4-3-0 : 1.3%
• 5-4-4-0 : 1.2%
• 5-5-3-0 : 0.9%
• 6-5-1-1 : 0.7%
• 6-5-2-0 : 0.65%
• 7-2-2-2 : 0.51%
• any other hand: less than 1.5%

LONGEST SUIT

Another way to think about distribution to consider how many cards are there likely to be in your longest suit.

If you've got 13 cards, at least one suit MUST contain four cards. However it's actually more likely that your longest suit will have five cards! Here are the figures for the chances of how long your longest suit will be:

• Four card suit: 35%
• Five card suit: 44%
• Six card suit: 16.5%
• Seven card suit: 3.5%
• Eight card suit: 0.47%
• Nine card suit: 0.037%
• Ten card suit: 0.0017%
• Eleven card suit: 0.000036%
• Twelve card suit: 0.00000032%
• Thirteen card suit: .....well, as the Duchess said to the colonel...

POKER HANDS

Poker games involve seeing what combinations you can get when you are dealt five cards.

The maths involved can be pretty murderous, but here are the chances for the combinations people hope to see, stating with the best possible hand - the Royal Flush.

Some people dream of this sort of thing

• 1 in 650,000 :ROYAL FLUSH (A,K,Q,J,10 all the same suit)
• 1 in 72,000 : STRAIGHT FLUSH (a run of five cards all in the same suit e.g. 7,8,9,10,J all hearts)
• 1 in 4,000 : FOUR OF A KIND
• 1 in 700 : FULL HOUSE (three of a kind AND a pair)
• 1 in 500 : FLUSH (all five cards the same suit)
• 1 in 256 : STRAIGHT (e.g. a run such as 2,3,4,5,6)
• 2% : THREE OF A KIND (e.g. three aces )
• 5% : TWO PAIRS (e.g. two eights and two threes)
• 42% : ONE PAIR (e.g. two queens )

If you haven't even got a pair, then whoever has the highest card wins. The WORST hand you can get is 2-3-4-5-7 of mixed suits.

Because the chances of a really good hand are quite small, a lot of poker games give you more than five cards, from which you can select which five cards you want to make the best combination. This lowers the odds of each hand, but they are still ranked in the same order as listed above e.g. a flush still beats two pairs.

TEXAS HOLD 'EM

Texas Hold 'Em is one of the most popular poker games. It's not just a matter of hoping to have the best hand, it's also a case of guessing what the other players might have. You might also have to guess if they are bluffing (i.e. betting when they've got nothing) and you can even try to bluff yourself. Here's what roughly what happens:

• Two of the players pay a little bit of money into the "pot". This money is usually called THE BLIND because nobody has seen any cards yet. The players take turns to do this, and it makes sure there's always a little bit of something to play for.
• Each player is dealt two cards that no one else sees.
• The players then take turns to see what they want to do. If a player likes his cards he can put in some more money. The other players can either "fold" which means drop out, "call" which means to put in the same money, or "raise" which means to put in the same money and then add some more! It keeps going until everybody who wants to keep playing has put in the same money. (Anybody who doesn't think it's worth it can drop out but they have to leave any money they have already bet in the pot.)
• Three more cards are dealt face up on the table for everybody to see. These three cards are called THE FLOP.
• There's more betting as before. (If nobody wants to bet they all "pass" and the next card is dealt...)
• A fourth card is dealt face up, this is sometimes called THE TURN.
• Even more betting until everyone is either happy or has folded.
• Finally the fifth card is dealt face up, known as THE RIVER.
• Anybody who is left playing then bets until just one person is left or they all agree to show their cards.

Quite often somebody wants to bet so much money that everybody else folds and he wins. In this case nobody gets to see anybody else's cards and the trouble is, you'll never be absolutely sure if the winner was bluffing or not!

We can't give you any help in how to bet apart from to remind you that nobody ever ruined their life by betting with matches or counters, so stick to playing with those and keep your money safe. What is far more interesting to us is how you make your five card hand.

Let's say that these are the five cards lying face up on the table

This player has got two TENS. There's also a third ten on the table, so the best five card hand the player can make is THREE TENS. This is quite a good hand, and the player might be tempted to bet a lot.

This player has got a SIX and a NINE. This looks disappointing, and most players would have thrown it away before the other cards were dealt onto the table. But it turns out to be very lucky because there is a SEVEN, EIGHT and TEN on the table. Therefore there is a five card straight leading up to the TEN. The player can feel very confident that this is the best hand. (In fact the only thing the other player could be holding to beat it is a nine and a jack. This would give a straight up to the Jack which is better.)

The "DEAD MANS HAND" is the nickname given to two pairs, the black aces and the black eights. In 1876 the gunfighter WILD BILL HICKOK was murdered in a saloon in Deadwood, Black Hills, Dakota while playing poker. Legend has it that these were the cards he was holding when he was shot. Nobody is sure what the fifth card was, or even if it had been dealt.

Back to "Do You Feel Lucky?"

Murderous Maths Main Index Page