Module 3 · Lesson 14 of 39 · Free
How often you are dealt a pair, aces or a suited hand
You are dealt a pair of aces once in 221 hands on average: at 30 hands an hour, about once every 7 hours. Every chance of a starting hand comes from one division, its combinations over all 1,326 starting hands. Aces have 6 of them, and 1,326 ÷ 6 = 221.
The hand
Your hole cards: A♠ A♥, the best starting hand in Hold'em. Enjoy them: you will not see them often.
Predict first
Your first card is dealt. What is the chance that your second card pairs it?
- 1 in 13: one rank out of 13
- 1 in 17: 3 of the 51 cards left
- 1 in 221: as rare as aces
Show the answer
Answer: 1 in 17: 3 of the 51 cards left
Whatever your first card, 3 of the 51 cards left share its rank: 3 ÷ 51 = 1 ÷ 17 ≈ 5.9%. Counting combinations gives the same: the 13 pairs make 78 of the 1,326 hands, and 78 ÷ 1,326 = 5.9%.
Working it out
Divide each hand's combinations by 1,326. Any pocket pair: 78 ÷ 1,326 = 5.9%, 1 hand in 17. One given pair, such as AA or 77: 6 ÷ 1,326 = 0.45%, 1 hand in 221. Two cards of the same suit: 312 ÷ 1,326 = 23.5%, almost 1 hand in 4. AK of any suits: 16 ÷ 1,326 = 1.2%, about 1 hand in 83, and AKs alone 4 ÷ 1,326 = 0.3%. At least one ace: 198 ÷ 1,326 = 14.9%, about 1 hand in 7.
Use these numbers to keep your expectations honest. In 100 hands expect about 100 × 78 ÷ 1,326 ≈ 5.9 pocket pairs, about 23.5 suited hands and less than half a pair of aces. Every pair comes equally often, deuces as often as aces, and so does every suited hand, AKs as often as 72s. Only the number of combinations makes a difference: you hold any offsuit kind three times as often as the same two ranks suited, 12 combinations against 4.
Formula
Chance of a hand = its combinations ÷ 1,326: aces 6 ÷ 1,326 = 1 in 221, any pair 78 ÷ 1,326 = 1 in 17
Check your understanding
How often are you dealt JJ, QQ, KK or AA, in %?
Show the answer
Answer: 1.81%
Four pairs of 6 combinations each: 4 × 6 = 24. Then 24 ÷ 1,326 = 1.81%, about 1 hand in 55.
On average, once in how many hands are you dealt AQo, an ace and a queen of different suits?
Show the answer
Answer: 110.5
AQo has 4 × 3 = 12 combinations: 1,326 ÷ 12 = 110.5 hands.
On your next hand, which is more likely: a pair of aces or a pair of twos?
- A pair of aces
- A pair of twos
- Both are equally likely
Show the answer
Answer: Both are equally likely
Each pair can be dealt in 6 ways, so each comes 6 times in 1,326, 1 hand in 221.