Module 3 · Lesson 16 of 39 · Free
What the flop brings: sets, flush draws and pairs
A pocket pair flops a set or better about once in 8.5 flops, 11.8% of the time. Every chance on the flop comes from counting flops: how many can come, and how many of them give you what you want. Often it is quicker to count the flops that miss.
The hand
Before the flop. You hold 7♣ 7♦, and the flop will be three of the 50 cards you have not seen.
Predict first
How many different flops can come?
- 117,600: 50 × 49 × 48
- 22,100: three of all 52 cards
- 19,600: three of your 50 unseen cards
Show the answer
Answer: 19,600: three of your 50 unseen cards
50 × 49 × 48 = 117,600 counts each flop 6 times, once for every order its three cards can come in: 117,600 ÷ 6 = 19,600. Your own two cards cannot be on the flop, so the flop comes from 50 cards, not 52.
Working it out
There are 50 × 49 × 48 ÷ 6 = 19,600 flops: the 6 counts the orders the same three cards can come in. With 7♣ 7♦, the flops with no seven take all three cards from the other 48: 48 × 47 × 46 ÷ 6 = 17,296. So a set or better comes 1 − 17,296 ÷ 19,600 = 11.8% of the time. With two hearts, 11 hearts are left, and 11 × 10 × 9 ÷ 6 = 165 flops are all hearts: a flush 0.84% of the time. Exactly two hearts, a flush draw, come in 11 × 10 ÷ 2 = 55 pairs of hearts, each with one of the 39 other cards: 55 × 39 = 2,145 flops, 10.9%. Two unpaired cards such as K♠ Q♦ have 6 cards that pair them, and 44 × 43 × 42 ÷ 6 = 13,244 flops have none of them: they pair at least one card 1 − 13,244 ÷ 19,600 = 32.4% of the time.
Most flops miss most hands, and that shapes the game. Two unpaired cards miss both their ranks on about two flops in three, and a pocket pair misses its set 88.2% of the time. So after the flop your opponent usually holds the same nothing as you, and the betting decides many pots that the cards do not. Module 6, After the flop, counts what the turn and the river can add.
Formula
Flops = C(50, 3) = 50 × 49 × 48 ÷ 6 = 19,600. At least one hit = 1 − misses ÷ 19,600: a set with a pocket pair 1 − 17,296 ÷ 19,600 = 11.8%
Check your understanding
You hold 9♠ 9♥. How often does the flop bring at least one card higher than a nine (a ten, jack, queen, king or ace), in %?
Show the answer
Answer: 79.3%
20 cards are higher than a nine. Flops with none of them come from the other 30: 30 × 29 × 28 ÷ 6 = 4,060. So 1 − 4,060 ÷ 19,600 = 79.3%.
You hold 5♣ 5♦. In how many of the 19,600 flops do both other fives come?
Show the answer
Answer: 48
Both fives must come, and the third card can be any of the other 48: 48 flops.
You hold A♣ 4♣. How often does the flop bring at least one more ace, in %?
Show the answer
Answer: 17.3%
3 aces are left. Flops with none of them come from the other 47: 47 × 46 × 45 ÷ 6 = 16,215. So 1 − 16,215 ÷ 19,600 = 17.3%.