Home » Articole » EN » Games » Gambling » Poker » Texas hold 'em probability » Flopping overcards when holding a pocket pair

# Flopping overcards when holding a pocket pair

It is useful to look at the chances different starting hands have of either improving on the flop, or of weakening on the flop. One interesting circumstance concerns pocket pairs. When holding a pocket pair, overcards (cards of higher rank than the pair) weaken the hand because of the potential that an overcard has paired a card in an opponent’s hand. The hand gets worse the more overcards there are on the board and the more opponents that are in the hand because the probability that one of the overcards has paired a hole card increases. To calculate the probability of no overcard, take the total number of outcomes without an overcard divided by the total number of outcomes.

Where x is the rank 3–K of the pocket pair (assigning values from 3–10 and J–K = 11–13), then the number of overcards is (14 – x) x 4 and the number of cards of rank x of less is 50 – (14-x) x 4 = 4x – 6. The number of outcomes without an overcard is the number of combinations that can be formed with the remaining cards, so the probability P of an overcard on the flop is

P = ((4x-6)/3)/(50/3),

and on the turn and river are

P = ((4x-6)/4)/(50/4) and P = ((4x-6)/5)/(50/5), respectively.

The following table gives the probability that no overcards will come on the flop, turn and river, for each of the pocket pairs from 3 to K.

Holding pocket pair Overcard on flop Overcard by turn Overcard by river
Prob. Odds Prob. Odds Prob. Odds
KK 0.7745 0.29 : 1 0.7086 0.41 : 1 0.6470 0.55 : 1
QQ 0.5857 0.71 : 1 0.4860 1.06 : 1 0.4015 1.49 : 1
JJ 0.4304 1.32 : 1 0.3205 2.12 : 1 0.2369 3.22 : 1
TT 0.3053 2.28 : 1 0.2014 3.97 : 1 0.1313 6.61 : 1
99 0.2071 3.83 : 1 0.1190 7.40 : 1 0.0673 13.87 : 1
88 0.1327 6.54 : 1 0.0649 14.40 : 1 0.0310 31.21 : 1
77 0.0786 11.73 : 1 0.0318 30.48 : 1 0.0124 79.46 : 1
66 0.0416 23.02 : 1 0.0133 74.26 : 1 0.0040 246.29 : 1
55 0.0186 52.85 : 1 0.0043 229.07 : 1 0.0009 1057.32 : 1
44 0.0061 162.33 : 1 0.0009 1095.67 : 1 0.0001 8406.78 : 1
33 0.0010 979.00 : 1 0.0001 15352.33 : 1 0.0000 353125.67 : 1

Notice that there is a better than 35% probability that an ace will come by the river if holding pocket kings, and with pocket queens, the odds are slightly in favor of an ace or a king coming by the turn, and a full 60% in favor of an overcard to the queen by the river. With pocket jacks, there’s only a 43% chance that an overcard won’t come on the flop and it’s better than 3:1 that an overcard will come by the river.