The Free-Card Myth

Many people assume that if they could only read their opponents perfectly, they'd be poker experts. I've long contended that many players wouldn't be experts even if they played with all the cards faceup.



Let's say you're playing no-limit hold'em and $50 goes into the pot preflop. At this point, all the cards get turned up. You have the K Q, and your opponent (damn him!) has two aces, including the A. You each have $40 left in your stack.



The flop comes down 10 9 2, giving you a gutshot-straight-flush draw. Your opponent, for some reason, checks. But he promises to put you in on the turn if he still has the best hand. What should you do?



You should take the free card, right? You both have perfect information, so you know your opponent isn't going to fold if you bet, and you've studied the odds and know you have only a 42 percent chance to win the hand, so you shouldn't value bet. OK, you check. The turn is the 2 and now the aces move all in. At this point, you're getting 2.25-1 pot odds and you're a 3.4-1 dog. You have to fold; oh well.



But wait, could you have done better? What happens if, instead of taking the free card on the flop, you move in? Now the aces call, and 42 percent of the time you win $90 (the $50 from the preflop action, and the $40 the aces put in on the flop); 58 percent of the time you lose $40. So, the expected value (EV) from your all-in bet is .42 x $90 – .58 x $40 = $14.60. Hey, you showed a profit! Well, of course you did, because if the aces had moved in on the flop, you would've been getting the right price to call, as you were much less than a 2-1 dog, and the pot was offering that same 2.25-1.



Still, was moving in with the worst hand really a better play than taking the free card? After all, checking the flop allows you to hit your hand on the turn for free. Let's calculate the EV of taking the free card. There are 45 cards that could come on the turn. On 11 of them (eight hearts and three non-heart jacks – the A is in your opponent's hand), the aces check and fold. On six of them, when you pair your king or queen, you will have picked up enough outs to be getting the right price against the aces' all-in bet – even with only one card to come. (Specifically, you'll have 16 outs with 44 possible river cards, making you a 1.75-1 dog while getting 2.25-1 on your money.) On the other 28 river cards, the aces shove in and you have to fold. So, the EV = (11/45) x $50 + (6/45) x ((16/44) x $90 – (28/44) x $40) + (28/45) x $0 = $13.19. That's $1.41 less than you make by moving in.



This analysis certainly seemed counterintuitive to me the first time I looked at it. The drawing hand is better off getting all of his money in with the worst hand than he is taking the free card? How can that be? Ah, but he knows his opponent will give him only one free card. If the choice were between moving in and checking it down, the straight-flush draw would obviously rather check it down. But his options are to check and see one more card, or put in $40 and see two more cards. And that changes everything. The made hand, in this case, would rather wait for a safe turn card before putting his money in, and the drawing hand would rather pay to see both the turn and river cards than get a free look at just one of them.



OK, Matt, but this is just some math problem, right? In real life, you always take free cards with a draw, and you never give free cards with a made hand – right?



Let me tell you about a hand I played in a $1,000 buy-in no-limit hold'em tournament. The blinds were $75-$150 and I opened for $400 from middle position with two red kings. Sam Grizzle called me from the big blind. Sam is a very good player who had been playing a highly loose/aggressive/tricky style.



The flop came down 9 6 5. Sam led out for $1,000 into this pot of $875. He had $6,900 behind after his bet, and I had him covered. Now, obviously Sam's cards weren't faceup, but I felt strongly that he had some kind of big draw, and possibly a pair to go with it. If my read was right and I raised, Sam would move all in. I would call, and would probably be a small underdog.



Instead of raising Sam's bet, however, I just called it. The turn brought the 4 – a card I liked. It didn't complete any of the reasonable straight draws, and it didn't bring a flush. Sam led out for $4,000 into what was now a $2,875 pot, leaving himself with $2,900. Obviously, he was pot-committed. But my read hadn't changed, and I set him all in. Sam called pretty quickly with the 7 5.



Doing some results-oriented analysis, if Sam and I had put all the chips in on the flop, I would've had a 45.4 percent chance of winning the hand. My EV for moving in on the flop was .454($16,675) – $7,900 = -$330. Yup, jamming with the kings here would've been a losing play against the hand Sam had.



To calculate my EV for just calling the flop, we have to make a few assumptions. First, I really do think I could've mucked the turn if a diamond had fallen. I'd have been in a tougher spot if a 7 or 8 had fallen. Let's say I'd get away from those half the time. If a 5 had come off, I would've gotten my chips in, and doubled Sam up if I failed to catch a lucky river. Sam, by the way, was very likely to pot-commit himself no matter what came on the turn. He had a great semibluffing hand, and aggression was his game.



It turns out that my EV from flat-calling the flop, given the turn strategy I just outlined, is +$880. (The calculation would take up too much space here, but I encourage you to try it.) Waiting for a safe turn card before I got aggressive made me about $1,200 in chip EV. Sam showed he was aware of this concept when he muttered after the hand was over, "I didn't much like it with one to come." Even if I pay off on some of the flush cards, my play was clearly better than getting it all in on the flop. Now you might say, "Of course you did better waiting for the turn, you weren't even a favorite on the flop!" That's true, but changing my flop equity from 45.4 percent to 50.1 percent makes the EV of jamming the flop only +$454. I still do better by waiting for the turn to be aggressive, assuming I have a good read.



For those who are curious, the river brought the J and Sam doubled up. Notice how insignificant that result was to this analysis.



Matt Matros is the author of The Making of a Poker Player, which is available at http://www.cardplayer.com/.