The maths of Jacks

	Pocket Jacks may well be the best hand before the flop. But more so than most hands, the flop can make you or break you

Poker is such a situation dependent game that the answer to a vast multitude of poker strategy questions is ‘it depends’. And if there was ever a poster child for that phrase, it would have to be pocket Jacks. At once a blessing and a curse, Jacks aren’t just called fishhooks because of their shape. They hook you in with their apparent strength, but often leave you dangling after the flop, and playing them is far from easy.

This is the first of a two-part series about playing wired Jacks. This month I’ll be exploring some of the mathematical parameters associated with this hand. Think of it as background information to give you an idea about the inherent strength of the hand, along with the statistical chances of your fishhooks surviving their first critical challenge – the appearance of a Queen, King or Ace on the flop. Next month I’ll examine how best to play pocket Jacks in fixed-limit and no-limit cash games, as well as in tournaments; but let’s look at the maths first.

Jacks are very flop-dependent

Although Jacks are tough enough to play in a limit cash game, a loss there can only amount to a few more chips. But playing them is much riskier in a tournament, particularly a no-limit tournament, where all your chips can be put at risk. The first thing to remember is that as with most hold’em hands, Jacks are flop-dependent as well as front-loaded.

Many players have an inflated opinion about just how strong pocket Jacks are and it’s easy to see why. Before the flop, the only hands better than a pair of Jacks are pocket Aces, Kings and Queens, and you’re not likely to run up against them very often. If you’re fortunate enough to be dealt a pair of Jacks it will probably be the best hand before the flop. But more so than most hands, the flop can make you or break you.

Simulate this

I used Mike Caro’s Poker Probe to run a series of cold simulations designed to provide some information about the intrinsic strength of a pair of Jacks. Then I used that information as a starting point to guide my thinking about how to best play pocket Jacks in a variety of common circumstances.

It’s important to understand that cold simulations are not real poker. Simulated players do not fold as players would in real games. Each hand is dealt to its conclusion and the result achieved through a cold simulation is really a power rating of sorts. Think of it as a numerical index of a hand’s raw strength and potential power. Tied hands are folded into winning percentages by Poker Probe, with the software allocating the entire pot for a win, half the pot for a two-way tie, one-third the pot for a three-way pot, all the way down to a nine-way tie.

SIMULATION 1: In the first simulation, there were nine players at the table, and I gave one of them J♥-J♣. Each simulation was run 500,000 times. I specified a board of 10♠-6♦-2♣, which minimised possible straights and flushes, and did not contain an overcard to the Jacks. With a situation this favourable, that pair of Jacks went on to win more than 20% of the confrontations, while each of the other eight random hands won approximately 10% of the time.

SIMULATION 2: But what if you’ve raised with a pair of Jacks and an overcard falls? Another simulation was run in which the board was Q♠-6♦-2♣. With an overcard on the board and the chances of a straight or flush minimised due to the distribution of suits and ranks, a pair of Jacks won only 14% of the confrontations. By contrast, the eight random hands won approximately 10.75% of the encounters. In other words, the pair of Jacks lost some equity because of the overcard, which benefited each of the other random hands. Nevertheless, a pair of Jacks was still inherently more powerful than any of the random hands, even when an overcard flops.

SIMULATION 3: In a third simulation two overcards appeared on the board and the random hands won more than 11% of the time, while the pair of Jacks won only 9.6% of the encounters, making a pair of Jacks a distinct underdog against a board containing two overcards.

SIMULATION 4: In the fourth and final simulation three overcards appeared on the board and the Jacks won nearly 13% of the time, compared to a little less than 11% for the random hands.

However, these figures are slightly misleading. The pocket Jacks didn’t really beat the random hands outright, but they did split the pot far more frequently. The message here is that a pair of Jacks is only slightly stronger than random cards when three overcards flop, and if you wind up making a straight with a pocket pair of Jacks, it stands a good chance of splitting the pot rather than winning it.

Summary of the results

JACKS, NO OVERCARDS FLOP

Wins 20.5% of the time; each of the eight other hands wins 9.9% of the time. JACKS, ONE OVERCARD FLOPS

Wins 14% of the time; each of the eight other hands wins 10.75% of the time. JACKS, TWO OVERCARDS FLOP

Wins 9.6% of the time; each of the eight other hands wins 11.25% of the time. JACKS, THREE OVERCARDS FLOP

Wins 12.9% of the time; each of the eight other hands wins nearly 11% of the time.

A couple of caveats…

In the real world, few players will call all bets until the river without a good hand. Because each pot is likely to have fewer than nine players, a pair of Jacks will win a higher percentage of hands than are shown in this cold simulation. In a cold simulation each and every hand calls to the river. In a real game players dealt hands like 7-2, 9-4, J-3 are not going to call the big blind, never mind calling a raise.

Even players with tempting hands, such as 9-8 or 10-9 are probably going to fold if the pot is raised before it’s their turn to act. That’s why most players raise before the flop with pocket Jacks. They want to eliminate as many opponents as possible thereby giving their pocket pair the best chance of winning.

Of course, any hand that calls the flop increases its chances of winning too. Nevertheless, in a fixed-limit cash game that pair of Jacks is probably going to be involved in the pot regardless of what transpires before the flop. The lesser hands really amount to a changing cast of characters – probably two or three opponents most of the time, not the aggregate eight other hands that are shown in this simulation.

As stated, Jacks are very dependent on the flop – and if you gain only one strategic insight from this article, it ought to be an awareness of how very flop-dependent this hand really is. And bear in mind that a pair of Jacks will be lucky to see a flop without an overcard.

In cold simulations, all overcards are equally dangerous as far as Jacks are concerned, but it’s quite different in the real word, since most of your opponents will play many more hands with an Ace in it than other big cards. Some players call all the time with A♣-7♠ and A♣-5♣, but you’ll seldom see your opponents turning up hands like K♣-6♠ and Q♣-5♣.

The implications should be crystal clear. If you hold pocket Jacks in a real poker game, be a lot more wary of a flop containing an Ace than one that contains a King or a Queen as its lone overcard. Better yet, if you can thin out the field with a raise before the flop, facing a one-overcard board isn’t nearly as daunting. When you’re heads-up or facing two opponents, chances that the lone overcard helped your opponents are less than they would be if you were involved in a family pot, where you can safely assume that any flop will help someone. If you look at a flop with two or three overcards, the situation is even worse than it appears from the simulated data. After all, anyone betting into a big board is likely to have part of it, and anyone calling a bet into this kind of board either has part of the flop or a draw to a big hand. While one overcard on the flop can be dangerous, two or three of them will cut your Jacks off at the knees.