Probability and Perceived Range in No-Limit Hold'em

We constantly ask ourselves what our opponent's range is, but our opponent is probably doing the same thing! So, in order to plan our plays effectively, we need to analyze what he thinks of our range, that is, our perceived range. If our perceived range is too strong, he will play cautiously, if it is too weak, he will be confident in playing his hands.

Perceived Range

Let's analyze how the villain perceives our range in an example.

Table setup

UTG: $895
Hero (CO): $1.752
BTN: $191
SB: $2.755
BB: $1.103

Blinds: $5/$10

Pot: $15

Our hand: :Qc :Jh

The hand starts with a fold from UTG and a raise from us to $35 from the CO. Since the villain thinks we are good and aggressive, they will assume that we have a wide range of hands. This means something like 36% of hands, or:

22+, A2s+, K3s+, Q6s+, J7s+, T7s+,
97s+, 87s, A4o+, K8o+, Q9o+, J9o+ and T9o

The button then folds and both blinds call the raise. The pot is now $105.

The flop comes: :Td :9s :5h

Party Poker is back in Brazil! Learn about the bonus and benefits when creating your Poker Dicas affiliate account.

Probabilities

The small blind chooses to check, an action he would do with most hands. Now, how does this flop affect our perceived range? Taking a quick look at our range as outlined above, it looks like we could hit the flop with many of the hands, but many hands like Ace-low or King-low missed the flop completely. If we look at the odds of hitting each hand, we get:

Considering the probability I calculated in another article on Continuation Betting, that the chance of the small blind having a hand with absolutely no value is about 8.8%, this means there is a 90% chance of him hitting something.

The small blind's range is much stronger than our perceived range. This means that, in general, the small blind will be confident about the strength of his hand.

Now that he has realized all of this, and he is probably also thinking about his own perceived range, it is time for him to act.

After the small blind checks, the big blind makes the same play and you bet $60 into a pot that now has $165.

Our c-bet is relatively small, the small blind should notice this and try to interpret its meaning. Why bet $60 here? Which hands would we bet more with?

He probably understands that we know that most of his and the big blind's hands include some sort of strong draw. So if we want to protect our hand against these draws, we would probably have to bet more, forcing them to fold. So the fact that we don't bet big gives the small blind a clue (a clue, not a certainty) that our hand is either a complete bluff, a draw, or an extremely strong hand, something like a set.

The fact that we bet even though we know his perceived range is strong could tell the small blind that our hand has good equity. Again, using the formula shown in the Continuation Bet article, the expected value of a $60 bet is $170.71.

$170.71 = (0.559)(0.541)($105) + {1- (0.559)(0.541)][(0.384)($675) – $60]

Most of this information, except for our equity when called and the probability of him folding, which is slightly different for him since he doesn't know we have QJ, is available to all players. So in theory, he could solve the above equation for an unknown range and calculate the equity that makes a bet of $60 more profitable than a larger bet. This would give him an idea of the strength of our hand and why our bet of $60 is the most profitable.

Of course, no one would do this kind of calculation during the game, let alone to try to figure out someone else's equity, but it's interesting to know that it's possible. With his calculations, the small blind probably deduces that we have at least some kind of hand on this flop, so he can narrow our range to something like:

55, 99-AA, A5o, A9o+, K9o+, Q9o+, J9o+, T9o,
A5s, A9s+, K5s, K9s+, Q8s+, J7s+, T7s+, 97s+ and 87s

Manipulating your perceived range

All of the above is correct when everyone is playing straight and simple. However, sometimes you will need to represent a different range than what you actually have. This is essentially the idea behind bluffing. However, there is also another type of bet that is related to bluffing. Let's call it overplaying your hand, which is the following two types of bets:

1. Distorting your perceived range
2. Overplaying a hand that is in your perceived range

For example, consider the following board:

:Ts :Tc :5d :Ad :9s

On this river, our opponent checked and we bet. Now, if we have played the hand extremely aggressively, we are representing a range with a lot of Tens and maybe a pair of Fives, but not with weaker hands. If, in fact, we do not have a Ten, but we do have AK, we are either misrepresenting our perceived range, or simply representing a hand that we do not have. Since our hand has some value, we expect someone who does not believe that we hold the hand we represent, but still has a hand that we can beat, to call our bet. In this case, we could be called by hands with weaker Aces or even pocket Kings, Queens or Jacks. This is an example of misrepresenting your range.

Take advantage of these 5 minutes of study and create your 888 Poker account by clicking this link! You'll get $88 free* and you can also play our weekly $100 freeroll every Tuesday! (*$8 in cash and $80 in bonus).

Now, consider the following board:

:Td :5s :4s :Ad :9s

If we have played our hand as if it had an Ace or a flush draw, and in fact we do have an Ace, our bet would be an overplay if we bet big on the river. Note, however, that when our bluffing hands are in our perceived range, it is easy for our opponent to see that we are bluffing. However, in the hand above, by betting big on the river, we are representing something completely different than what our hand actually is, making it much harder for our opponent to figure out what our hand is. This is an example of overplaying a hand.

Now, let's look at this board:

:5d :4h :3s :Jd :As

If we have slowplayed a straight on the flop, either by checking or just calling, our perceived range will be weaker than our hand, allowing us to bet for value and get called by worse hands. This is another example of misrepresenting your range. However, by slowplaying, we are pretending to demonstrate a weaker range than we actually have, but we still expect, as in the first example, that our opponent will call our bet. The difference is that now we beat virtually any hand that calls us, whereas in the first example this was certainly not the case.

Article translated and adapted from the original: Classic Book Excerpt: Analytical No-Limit Hold 'em – Perceived Ranges