Understanding the Probability of Forming a Pair in Poker

The probability of flopping a Pair when starting with two unpaired cards stands at 29%, translating to 1 in 3.45.

Pair – Quick Recap

Definition of a One Pair –

Holding two cards of identical rank.

Example – AAXXX

In Texas Hold’em, a Pair is defined as holding two cards of the same rank in your initial hand. It can also occur when one of your hole cards matches a card on the board, or when there are Pairs among the community cards. The strongest Pair possible in Hold’em is Aces .

Odds of Making a Pair on the Flop

The likelihood of forming a Pair on the flop is influenced by the specific hand you possess.

If you are holding two unpaired cards, the probability of getting a Pair on the flop is 29% .

Nonetheless, the overall probability of forming a Pair with any combination of starting cards is actually 31.1%.

So, why the discrepancy?

Of course, some of the hands we start with may already include Pocket Pairs! This guarantees that we can make at least a Pair or better. Even if we disregard any Pairs on the board (in such scenarios, we still regard our hand as One Pair), the statistics for Pocket Pairs are as follows.

The Pocket Pairs - Breakdown

A Weak Pair (below the second highest card on the board) = 41.3%

A Mid Pair (above the second highest card on the board) = 23.3%

Over Pair (above top card on the board) = 23.3%

Trips and better = 12%

The probability of different Pair types varies based on the specific Pocket Pair in your hand, as stronger Pocket Pairs are more likely to produce Over Pairs.

For example -

KK – Flops an Over Pair 67.3% of the time.

JJ – Flops an Over Pair 36.2% of the time.

88 – Flops an Over Pair 10.2% of the time.

Unpaired Holdings – Breakdown

Bottom Pair = 8.98%

Middle Pair = 10%

Top Pair = 10%

Once again, this is heavily contingent on the hand you start with.

Odds with AKo

Bottom Pair = 0%

Middle Pair = 0%

Top Pair = 29%

Odds with 98o

Bottom Pair = 4.9%

Middle Pair = 15.6%

Top Pair = 8.45%

Odds with 32o

Bottom Pair = 26.9%

Middle Pair = 2.02%

Top Pair = 0%

It's hardly surprising that high-card hands tend to result in top Pairs more often than mid-range or low-card hands, which are more likely to form Mid or low Pairs, respectively.

For instance, AKo will never flop either a second Pair or a bottom Pair—think about that!

Probabilities of Forming a Pair in Later Rounds

Now, let’s picture a scenario where we completely miss the flop with an unpaired hand and are curious about our chances of drawing a Pair on either the turn or the river.

By this point, five cards are already visible (the flop along with your two hole cards). There remain 47 cards in the deck, six of which would give us a Pair .

Thus, the likelihood of forming a Pair on the turn is calculated as 6/47 = 0.12767 or approximately 12.8%.

As we transition from the turn to the river, there's one less card remaining in the deck, which meansthe probability of hitting a Pair on the river, after holding an unpaired hand on the turn, becomes 6/46 = 0.1304 or roughly about 13%

It's noteworthy that the odds are slightly improved when going from turn to river due to the reduced number of available cards.

Finally, let's assess our chances of making a Pair from the flop through to the river. We compute this by determining our chances of not forming a Pair and subtracting that from 100%.

The chance of not forming a Pair on the turn = 41/47

The chance of not forming a Pair on the river = 40/46

Therefore, the probability of not hitting a Pair on both streets is (41/47) * (40/46) = 0.7586, which is about 76%

Consequently, the chance of hitting a Pair from the flop through to the river is (100% - 76%) approximately 24%.

Implied Odds Analysis of a Pair

Obtaining a One Pair hand doesn't inherently ensure favorable implied odds. There are numerous scenarios in which opponents can improve on later streets, so we shouldn't assume large payouts simply from hitting.

On the flip side, one-pair type hands (particularly the lower Pairs) tend to generate reverse implied odds . Even if we make a Pair, there's still a significant chance of losing chips on subsequent streets if our opponent improves.

The silver lining is that one-pair hands (when correlated with the board) possess five outs to potentially form either Two Pair or Trips, and such holdings tend to perform decently in terms of implied odds.

Pocket Pairs have limited potential for improvement, but they occasionally result in strong yet concealed hands (like sets) which come with great implied odds.

Basic Strategy Advice

Exercise caution with one Pair, especially in larger stacks of 100bb or more. These hands can be easily outdrawn and are generally suited for small to medium-sized pots.

Exceptions to this include high top Pairs with strong kickers and over-Pairs, which may sometimes be sufficient for stack plays, contingent upon the specific circumstances.