How to Compute the Probability of No Pair in Stud Poker

In this lesson, we explain how to compute the probability of being dealt a hand with no pairs. And we will show how to compute probability of the weakest hand in stud poker - a hand with no pairs that is also not a straight, a flush, or a straight flush. (For a brief description of stud poker, click here.)

Hands With No Pairs

In stud poker, a pair refers to two cards of equal rank. There are four types of hands that do not have at least two cards of equal rank. That is, they do not have at least one pair.

• Straight flush. Five cards of the same suit in sequence, such as 3♥, 4♥, 5♥, 6♥, 7♥.
• Flush. Five cards of the same suit with at least one card out of sequence, such as 3♦, 4♦, 5♦, 6♦, K♦.
• Straight. Five cards in sequence, with at least two cards of different suits. Ace can be high or low, but not both. Thus, A♠, 2♥, 3♦, 4♣, 5♥ and 10♠, J♥, Q♦, K♣, A♥ are valid straights; but Q♠, K♥, A♦, 2♣, 3♥ is not.
• High card. Five cards of different rank with at least two different suits, such as 3♥, 5♥, Q♦,K♣, A♥.

How to Compute Poker Probabilities

In a previous lesson, we explained how to compute probability for any type of poker hand. For convenience, here is a brief review:

• Count the number of possible five-card hands that can be dealt from a standard deck of 52 cards
• Count the number of ways that a particular type of poker hand can occur
• The probability of being dealt any particular type of hand is equal to the number of ways it can occur divided by the total number of possible five-card hands.

So, how do we count the number of ways that different types of poker hands can occur? We recognize that every poker hand consists of five cards, and the order in which cards are arranged does not matter. When you talk about all the possible ways to count a set of objects without regard to order, you are talking about counting combinations. Luckily, we have a formula to do that:

In summary, we use the combination formula to count (a) the number of possible five-card hands and (b) the number of ways a particular type of hand can be dealt. To find probability, we divide the latter by the former.

Probability of No Pairs

Let's execute the analytical plan described above to find the probability that a randomly-dealt poker hand will not have any pairs.

• First, we count the number of five-card hands that can be dealt from a standard deck of 52 cards. This is a combination problem. The number of combinations is n! / r!(n - r)!. We have 52 cards in the deck so n = 52. And we want to arrange them in unordered groups of 5, so r = 5. Thus, the number of combinations is:

  ₅₂C₅ = 52! / 5!(52 - 5)! = 52! / 5!47! = 2,598,960

  Hence, there are 2,598,960 distinct poker hands.
• Next, we count the number of ways that five cards can be dealt to produce a hand that does not have any pairs. It requires six independent choices to produce a hand without any pairs:
  • Choose the rank of each card in the hand. A playing card can have a rank of 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king, or ace. Since there are no pairs (i,e., none of the cards have equal rank), we choose 5 ranks from a set of 13 ranks. The number of ways to do this is ₁₃C₅.
  • Choose a suit for the first card. There are four suits, from which we choose one. The number of ways to do this is ₄C₁.
  • Choose a suit for the second card. There are four suits, from which we choose one. The number of ways to do this is ₄C₁.
  • Choose a suit for the third card. There are four suits, from which we choose one. The number of ways to do this is ₄C₁.
  • Choose a suit for the fourth card. There are four suits, from which we choose one. The number of ways to do this is ₄C₁.
  • Choose a suit for the fifth card. There are four suits, from which we choose one. The number of ways to do this is ₄C₁.

  The number of ways to produce a hand with no pairs (Num_np) is equal to the product of the number of ways to make each independent choice. Therefore,

  Num_np = ₁₃C₅ * ₄C₁ * ₄C₁ * ₄C₁ * ₄C₁ * ₄C₁

  Num_np = 1287 * 4 * 4 * 4 * 4 * 4 = 1,317,888

  Conclusion: There are 1,317,888 different ways to deal a poker hand that has no pairs.
• Finally, we compute the probability. There are 2,598,960 unique poker hands. Of those, 1,317,888 have no pairs. Therefore, the probability of being dealt a hand with no pairs (P_np) is:

  P_np = 1,317,888 / 2,598,960 = 0.5070828331

Bottom line: In stud poker, players get a hand without any pairs on about half the deals.

Probability of a High Card Hand

The Venn diagram below shows the relationship between the various types of poker hand that do not have at least one pair. The rectangle represents the sample space of all poker hands. The white space represents poker hands that have at least one pair. Each circle is a particular type of poker hand that does not have a pair.

None of the circles overlap or intersect. This indicates that the various poker hands without a pair are mutually exclusive events. Therefore,

P_np = P_sf + P_s + P_f + P_hc

where P_np is the probability a hand does not have a pair, P_sf is the probability of a straight flush, and P_s is the probability of a straight, P_f is the probability of a flush, and P_hc is the probability of high card hand. To compute the probability of a high card hand, we rearrange terms, as shown below:

P_hc = P_np - ( P_sf + P_s + P_f )

We actually know the values of the probabilities on the right side of the equation. We computed P_np in the previous section on this web page. And we computed the other probabilities in earlier lessons. The table below shows probabilities for each term, along with a link to the relevant computations.

Term	Probability	Link
Pnp	0.5070828331	See computation
Psf	0.00001539077169	See computation
Ps	0.003924646781	See computation
Pf	0.001965401545	See computation

Given these probabilities, it is easy to compute the probability of a high card hand.

P_hc = P_np - ( P_sf + P_s + P_f )

P_hc = 0.5070828331 - ( 0.00001539077169 + 0.003924646781 + 0.001965401545 )

P_hc = 0.501177394

In stud poker, a high card hand is the weakest hand a player can get. Sadly, it is the hand that players get most often. About half the hands that are dealt turn out to be high card hands - no pairs, no straights, and no flushes.