As a mathematician, I find poker an interesting and complex game to study. Unlike blackjack, where card counting can give you a provable edge over the casino, poker is more subtle from the point of view of advantage gambling. It's you against the other players, not the casino, and optimal play is subjective rather than absolute.

This does not mean poker can't be studied statistically and your game improved. There are countless good books out there by well-known experts (including our own BJI writer Bill Burton of course!). With time and research, you can learn a number of strategies to win at poker.

As Bill points out in his article, Poker Drill Master is a different kind of training tool. Rather than playing poker, the program presents Texas Hold'em games in partially completed states with a bet to you. You have to answer a bunch of questions, ultimately deciding whether to fold, bet or raise.

To determine the right answer, Poker Drill Master uses a built-in poker evaluation engine. This high performance simulator completes the game in front of you millions of times using random draws, to come up with the win/loss statistics. Once the program has approximated your equity (probability of winning factoring in ties), the rest is easy: see if the money in the pot, relative to the bet you have to call, is sufficiently large enough to make up for the minimum required pot odds (based on your hand's equity).

So Poker Drill Master is not about strategies or studying your opponents. The idea is to quickly develop your intuitive skills at deducing pot odds through efficient repetition and practice.

And it's through these random drills that I have learned how surprising and unintuitive poker can be! Although you may often be fairly close on your guesses for the equity, once in a while a hand will completely catch you off guard. While the reports in Poker Drill Master include a wealth of information to help understand the reasons (including possible outs and the equities for the opponents), I want to explore some typical examples in this article.

Let's say you're in a traditional Texas Hold'em game as follows:

• The board: 7s Qh 9d Ad

• Your hand: 6s 6c
• One opponent (heads up play).
• $5 bet to you, $10 in the pot.

You figure with a terrible under pair, you should fold. But let's analyze the hand using equity and pot odds.

Using Calculator Mode in Poker Drill Master, we can enter this exact game and let the evaluation engine do its thing. In this case, there is only one card left in the board and two unknown opponent cards, so the program cycles through all 45,540 possible permutations, to yield the exact result:

Surprisingly, even with a low pair, we have about a 50/50 chance of winning. This means we need a minimum pot size of $5 on average (for a $5 bet), to profit from this hand. We're getting 2 to 1 odds ($10 to $5), so based only on the pot odds, we should call the $5 bet instead of folding.

Pot odds are not the only factors you should consider when betting. You might have an understanding of your opponents' playing styles, factor in bluffing, or be able to guess your opponent's hand from his initial bets this round. But pot odds (especially after the flop) are still the best foundation for low limit Texas Hold'em games, and you should deviate with caution!

What happens if there are more opponents in the round? This is where your intuition might fail you, since the actual relationship is not linear, but usually exponential. This means that as we add opponents, our equity drops by increasingly larger amounts (exponentially). An approximate equation that fits many poker hands (but not all… see below) is 'p^n', where 'p' is the equity against one player, and 'n' is the number of opponents:

We go from a 50-50 chance with one opponent, to a 1 in 13 long shot with only two extra players. Marginal hands can be playable head-to-head, but quickly become dogs with even a few extra opponents.

Consider the following game:

• The board: Td Jd 8c
• Your hand: Ks Qc

• One opponent (heads up play)

With two cards to go, you still have 8 outs for a straight (the four aces and four nines), so you might correctly predict that you have a bit better chance than your heads-up opponent of winning. You actual equity is 58.36% in this case, validating your instincts. But what happens as we add more players? With the above example showing the exponential relationship, we might think that our chance of winning is not good with 3 or more opponents. But the 'p^n' equation sometimes is very wrong for particular kinds of hands, including this one:

So this hand is quite playable, regardless of the number of players (assuming there is sufficient money in the pot to make the odds). Since the potential winning straight hand is so strong, it doesn't really matter as much when we add more opponents. This case definitely shows how complicated it can be to evaluate the strength of your hand.

These are only a couple small examples of surprising hands that can come up in Texas Hold'em. While most seasoned veterans would have guessed the right answers above, such knowledge only comes with experience when the results are counter intuitive.

Poker Drill Master is available for a special reduced price of $49.95 ($10 off) for BJI subcribers. Click here to obtain this special discount, and download the software.

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