Module 2 · Lesson 9 of 39 · Free
Playing the board
On a board of 9 8 7 6 5, every hand without a ten has the same best five: the straight on the board. Of the 1,081 two-card hands a player could hold there, 903 have no ten, and all of them tie. When the five board cards are your best hand, you are playing the board, and your own two cards add nothing.
The hand
The river: 9♠ 8♥ 7♦ 6♣ 5♠. You hold K♣ K♦, the Calling station shows A♥ 2♣, and there are 40 bb in the pot.
Predict first
Who wins the 40 bb pot?
- Nobody alone: you split it, 20 bb each
- You do: a pair of kings beats ace high
- He does: his ace is the highest card
Show the answer
Answer: Nobody alone: you split it, 20 bb each
The board alone is a straight, 9 8 7 6 5, and it is the best five for both of you: a pair of kings and an ace are worse than a straight, so neither plays. You both play the board, and the pot is split: 40 ÷ 2 = 20 bb each.
Working it out
Count the hands on 9♠ 8♥ 7♦ 6♣ 5♠. Besides the five board cards there are 47 others, and a player can hold any two of them: 47 × 46 ÷ 2 = 1,081 hands, halved because the same two cards can come in either order. Only a ten improves this board, to T 9 8 7 6, or with a jack too to J T 9 8 7, and no flush can be made, because no three board cards share a suit. The 47 cards hold 4 tens, so 43 are not tens, and 43 × 42 ÷ 2 = 903 hands have no ten. That makes 903 ÷ 1,081 = 83.5% of all hands playing the board, while the other 1,081 − 903 = 178 hold a ten and beat it.
A hole card still counts whenever it makes a better five than the board. On Q♥ 9♥ 7♥ 4♥ 3♥ any heart above the 3♥ plays: the J♥ makes Q J 9 7 4 of hearts, but the 2♥ is lower than every heart on the board and does not. On a board with two pair, a card above the fifth board card plays as the kicker. Playing the board ties you only with the players who play it too, and anyone with a better five takes the whole pot. When the board is the best hand possible, such as A K Q J T with no three cards of one suit, nobody can beat it and every player still in splits the pot.
Formula
On 9 8 7 6 5 of mixed suits, hands that play the board = the cards that cannot improve it, taken two at a time: 43 × 42 ÷ 2 = 903 of the 1,081 hands
Check your understanding
The board is A♠ A♦ K♣ K♥ Q♠ and you hold J♥ J♣. What is your best five?
- A♠ A♦ K♣ K♥ Q♠: the board
- A♠ A♦ K♣ K♥ J♥: your jack as the kicker
- A♠ A♦ J♥ J♣ K♣: your jacks with the aces
Show the answer
Answer: A♠ A♦ K♣ K♥ Q♠: the board
Aces and kings fill four places, and the fifth goes to the highest card left: the board's Q♠ beats your jacks. Aces and jacks would be worse than aces and kings, so your cards add nothing and you play the board.
The board is T♠ 9♥ 8♦ 7♣ 6♠. You hold A♠ A♥ and he holds J♦ 2♣. Who wins the pot?
- Nobody alone: you both play the board
- You do: you have a pair of aces
- He does: his jack makes a higher straight
Show the answer
Answer: He does: his jack makes a higher straight
The board is a straight to the ten, and a pair of aces cannot beat it, so you play the board. His J♦ makes J T 9 8 7, a straight to the jack, and he takes the whole pot.
The board is K♣ J♣ 8♣ 5♣ 3♣. Of the 1,081 two-card hands a player could hold, how many play the board?
Show the answer
Answer: 780
Eight clubs are left, A Q T 9 7 6 4 2, and all of them but the 2♣ are above the 3♣: 7 clubs improve the flush. The other 47 − 7 = 40 cards do not, so 40 × 39 ÷ 2 = 780 hands play the board.