Poker Hands Probability – Texas hold’em

 Poker Hands ProbabilityIn poker, the probability of many events can be determined by direct calculation.

When calculating poker hands probabilities for a card game such as Texas Hold ’em, there are two basic approaches. The first approach is to determine the number of outcomes that satisfy the condition being evaluated and divide this by the total number of possible outcomes.

The second approach is to use conditional probabilities, or in more complex situations, a decision tree.

When working out hand probabilities, the main probabilities we will work with are the number of cards in the deck and the number of cards we want to be dealt. So for example, if we were going to deal out 1 card:

  • The probability of dealing a 7© would be1/52– There is one 7© in a deck of 52 cards.
  • The probability of dealing any Ace would be4/52– There four Aces in a deck of 52 cards.
  • The probability of dealing any¨would be 13/52 – There are 13¨ s in a deck of 52 cards.

In fact, the probability of being dealt any random card (not just the 7© ) would be 1/52. This also applies to the probability being dealt any random value of card like Kings, tens, fours, whatever (4/52) and the probability of being dealt any random suit (13/52).

The numbers change for future cards.

  • The probability of being dealt a 7 for the first card will be4/52.
  • The probability of being dealt a 7 for the second card will be3/51.

After we have been dealt the first card, there is now 1 less card in the deck making it 51 cards in total. Also, after already being dealt a 7, there are now only three 7s left in the deck.

Probability of being dealt two exact cards.

Multiply the two probabilities together.

So, we want to find the probability of being dealt the Aª and K«. (See the “and” there?)

  • Probability of being dealt Aª- 1/52.
  • Probability of being dealt K«- 1/51.

Now let’s just multiply these bad boys together.

P = (1/52) * (1/51)
P = 1/2652

So the probability of being dealt the Aª  and then K« is 1/2652.

Probability of being dealt a certain hand.

Multiply the two probabilities together.

So, we want to find the probability of being dealt any Ace and any King.

  • Probability of being dealt any Ace –4/52.
  • Probability of being dealt any King –4/51(after we’ve been dealt our Ace, there are now 51 cards left).

P = (4/52) * (4/51)
P = 16/2652 = 1/166

However, again with the 2652 number we are working out the probability of being deal an Ace and then a King. If we want the probability of being dealt either in any order, there are two possible ways to make this AK combination so we multiply the probability by 2.

P = 16/2652 * 2
P = 32/2652
P = 1/83

Probability of being dealt a range of hands.

Work out each individual hand probability and add them together.

What’s the probability of being dealt AA or KK? (Spot the “or” there? – Time to add.)

  • Probability of being dealt AA –1/221(4/52 * 3/51 = 1/221).
  • Probability of being dealt KK –1/221(4/52 * 3/51 = 1/221).

P = (1/221) + (1/221)
P = 2/221 = 1/110

 

Some Poker Odds and Statistics -Poker Hands Probability

 

Premium hands

 

The chances of getting a top starting hand (of double aces, picture pairs or A-K suited), is a minute 2.1%.

Flush

 

If you’ve got a flush draw (one card short of a full flush) after the flop, you’ll make your hand 34.97% of the time.

 

Suited

 

Don’t play any two cards just because they’re suited. It only improves your hand by 2.5%.

 

Paired up

 

The chance of one of your hole cards making a pair on the flop is 32.43%, or about a third.

 

Hitting the board

 

By the river, your chances of making a pair go up to roughly a half.

 

Straight

 

If you flop an open-ended straight draw this gives you eight outs (eight possible cards that will complete the hand), so you’ll hit your hand by the river 31.5% of the time

 

Three of a kind

 

If you already have a pair, the odds of flopping a set (three of a kind) are only 7.5/1

 

 Inside straight

 

Rarely worth drawing to, with the turn and river cards to come you’ll hit your gutshot straight (four outs) approximately 9% of the time.

 

Over-pair

When two pairs go head to head, the bigger pair will win roughly 80% of the time (or four times out of five).

 

Perfect cards

If you need two exact cards on the turn and river, the chance of getting them both is only 0.3%. If you get one, there’s a 4.55% chance of getting the other.

 

 

 

 

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