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Drawing Across Multiple Betting Rounds

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By Tony Guerrera

When playing any variant of poker, making a decision on a betting round without accounting for future betting rounds is a fatal mistake. Avoiding this mistake while luring opponents into making it will contribute substantially to your bottom line.

Mistakes having to do with improperly accounting for future betting rounds occur often in drawing situations. Below are some enlightening examples.

Example #1: You’re playing 6-handed $1-$2 no-limit hold’em. UTG+1 and the button limp, SB folds, and you check your option in the big blind with Ts9s. The pot is $7 going to the flop, and the flop is Jd5s2s. You check, UTG+1 bets $4, and the button folds. Suppose that if you call and miss your flush, UTG+1 will bet $15 on the turn. Further suppose that UTG+1 won’t put any chips in the pot if a spade falls (you have no implied odds). What’s your EV if you take the passive line of play (calling and trying to hit the flush)?

On the flop, with two cards to come, you’ll hit your flush about 35% of the time. With two cards to come, you need pot odds of at least $65:$35 ≈ $1.86:$1 to call. UTG+1′s $4 bet on the flop gives you $11:$4 = $2.75:$1 odds to call. Initially, it looks like you’re getting the right pot odds to call. However, you know that if you miss your flush on the turn, your opponent will make a pot-sized bet of $15. To draw to your flush on the turn, you’d need $37:$9 ≈ $4.11:$1 odds to call. But a pot-sized bet of $15 would only give you $2:$1 pot odds. You’d have to fold your flush draw on the turn; therefore, you’re drawing with only one card to come.

Since you’re drawing with only one card to come, you need $38:$9 ≈ $4.22:$1 pot odds to call the $4 bet on the flop. You’re only getting $11:$4 = $2.75:$1 odds, so you should fold. The moral: when you’re drawing on the flop, you have to be very mindful of whether you’re drawing with 1 or 2 cards to come.

Since it’s not possible for you to draw profitably without implied odds, let’s figure out what implied odds are necessary to make the passive hit-to-win line of play in example #1 profitable. Letting x denote the additional amount of money that the big blind needs to win to make drawing profitable:

If you call on the flop, you need to get at least $5.89 from your opponent when you hit your flush, on average, to make drawing passively a profitable line of play in the long run. After calling on the flop, the pot will be $15. If you can get a $10 bet paid off more than 58.9% of the time, drawing will be profitable. If you can get an opponent to call a $7.50 bet about 78.5% of the time, the passive drawing line of play can be profitable.

To figure out whether you can expect the required implied odds, let’s now think of this hand from UTG+1′s perspective. The preceding discussion suggests that UTG+1 should avoid calling anything but very small bets when suspected draws hit. Not putting any money into pots will allow players to bluff him when phantom outs hit, but remember that players need to hit phantom outs in order to bluff them. Even if UTG+1 is playing an opponent who bluffs phantom outs 100% of the time, he’s not leaving himself open to exploitation as long as he denies opponents proper odds to hit their real and phantom outs in the first place with proper bet-sizing. If UTG+1 properly sizes his bets and never yields sufficient implied odds, the only way that draws can become profitable against him is if they’re played aggressively (i.e. semibluffs are essential).

Example #2: The situation is similar to example #1 with some minor, but interesting changes. You’re playing 6-handed $1-$2 no-limit hold’em. UTG+1 and the button limp, SB folds, and you check your option in the big blind with Ts9s. The pot is $7 going to the flop, and the flop is Jd5s2s. UTG+1 bets, and the button folds. Suppose you know that UTG+1 will make a bet on the turn that gives you the required $37:$9 odds to call (i.e you’re drawing with two cards to come). What’s the minimum bet from UTG+1 on the flop that makes a passive drawing line of play unprofitable for you if UTG+1 yields no implied odds?

The EV of the passive line of play is found by considering the payouts and probabilities associated with each outcome. Let F represent the size of UTG+1′s bet on the flop and T represent the size of UTG+1′s bet on the turn.

Outcome #1: You hit your flush on the turn

Probability =

Payout = 7 + F

Outcome #2: You miss your flush on the turn and hit your flush on the river

Probability =

Payout = 7 + F + T

Outcome #3: You miss your flush on the turn and miss your flush on the river

Probability =

Payout =

Next, we need to figure out how to express T in terms of F. If you call on the flop, the pot will be 7 + 2F on the turn. The odds you get on the turn are (7 + 2F + T):T.

Substituting for T in the expressions for the payouts, we can use the following inequality to solve for the minimum bet on the flop that makes the passive drawing line of play -EV (assuming no implied odds):

If the person defending against the flush draw makes a bet larger than $2.17 on the flop followed by a bet of $3.65 on the turn (the turn bet yielding $37:$9 odds), the player on the flush draw can’t profit if he plays his draw passively against an opponent who won’t yield implied odds. The $2.17 bet on the flop is only 31% of the pot, and the $3.65 bet on the turn is only 32% of the pot. These numbers suggest that, under many circumstances, draws should be played aggressively or not at all. Especially interesting is that the flop bet gives you $4.23:$1 pot-odds. And even though the odds against you hitting with two cards to come are only 1.86:1, it’s still improper to play the draw passively against an opponent who yields no implied odds!

Meanwhile, from UTG+1′s perspective, these numbers suggest that he doesn’t need to bet as much as is typically advocated to properly defend against the drawing portion of an opponent’s distribution. However, he should still bet more than 31% of the pot. First, a bet that’s less gives his opponent a +EV draw even with just one card to come. Second, sophisticated opponents will draw to real outs plus phantom outs. Third, opponents with made hands not as good as yours will be willing to call bets on the order of 50%-66% of the pot; betting just around 30% of the pot won’t extract the most possible value from them.

Some additional layers of strategic sophistication are needed to make this analysis of play involving drawing hands completely comprehensive. In particular, players’ hand distributions need to be carefully considered in their entirety. But we were still able to draw some interesting conclusions regarding drawing and defending against draws in no-limit hold’em:

  • Against opponents who don’t yield implied odds, it’s tough to play drawing hands passively in a straight-up hit-to-win fashion.

  • Semi-bluffing and phantom outs are important components of playing drawing hands profitably

  • The player defending against a draw doesn’t need to make very big bets on the flop or the turn to deny drawing opponents proper odds-really, they just need to be of a size that denies proper odds with one card to come.

  • The lower the defending player makes his bets, the lower the size of bets he can call in future rounds when scare card fall. Vice versa, for given stack depths, there exists a minimum bet size that leaves the defending player committed in future betting rounds (i.e even if a drawing opponent plays his hand face up and only bets when he hits his draw in a future betting round, the defending play can call and still profit from the overall line of play in the long run).

Tony Guerrera is the author of Killer Poker By The Numbers and co-author of Killer Poker Shorthanded (with John Vorhaus)

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