Hand Third Fourth Fifth Sixth
street street street street
Q♣-6♥-Q♦ 64 67 71 76
8♣-5♠-8♦ 36 33 29 24
Let us go back to the contest between a pair of Queens and a pair of Eights whose kicker is of a lower rank than that of the Queen. As you know, the small pair is 9:5 against in the third street. Its fortunes decline as the contest progresses to the higher streets. Thus, the smaller pair becomes 2:1 against at the fourth street, 7:3 against at card five and 3:1 against on the turn of the river card. You love pot-limit players who gamble with their small pairs to the river. They hate their money because they are getting one to one for their money with odds of, at best, just under 2:1 against.
The probability of improvement of each hand, on a card-by-card basis, will be discussed next.
2.1.1 Fourth street
Assume that you raised on third street when the dealer gave you Q ♣-6♥-Q♦. Only John, holding 8♣-5♠-8♦, calls your bet. Let us also assume that none of the seen discards included the cards needed by you and your rival. Each one of you will hit trips or two pairs 4% and 6% of the time on the turn of the fourth card. That means that out of every
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100 hands only John will improve to trips or two pairs about 10% of the time (9:1 against) on the fourth street. If both of you pair your door cards, John will be about 7:2 against; I am assuming he is sensible and would not contemplate such a gamble, especially in a pot-limit game.
Let us study your opponent’s opportunities to make money if he is the only player who gets the miracle cards on the fourth street (21:1 against).
Player Hand % win-rate
You (pair) Q♣-6♥-Q♦-4♠ 11.2
John (trips) 8♣-5♠-8♦-8♥ 88.8
When the dealer pairs John’s first upcard, you should give up the fight on most occasions because your big pair is 8:1 against. The exceptions to this rule are twofold:
(1) John may be the type of opponent who gambles with three flushing/straightening cards against a big pair.
(2) The discards may suggest otherwise—for instance, two Eights were discarded.
Furthermore, only John will have two pairs about 6% of the time. But he will lose 2.5 out of the 6 times in a hundred the dealer gives him, say, the 5♥ because you can still win about 45% of pots if you decide to gamble against his two pairs (6 × 0.55 ≅ 3.5)
Player Hand % win-rate
You (pair) Q♣-6♥-Q♦-4♠ 45
John (two pairs) 8♣-5♠-8♦-5♥ 55
Thus, if John’s hand is the only one which improves on the fourth street, he should expect to win about 7.5% (3.5 + 4) of the pots (≅12:1 against).
2.1.2 Fifth street
If your opponent loves his hand and decides to see the fifth card with you,
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then both of you will make trips or two pairs about 9% and 20% of the time respectively. Therefore, John will be the only player with trips about 8% of the time (13:2 against) in the fifth street; you will be worse than 12:1 against when he captures the third Eight.
Player Hand % win-rate
You (pair) Q♣-6♥-Q♦-4♠-9♦ 7.5
John (trips) 8♣-5♠-8♦-3♥-8♥ 92.5
He will also make two pairs, while you will receive blanks, 13% of the time by the fifth street. But since you will win 42% of the pots when you decide to gamble against his two pairs, he will win 7.5 (13 × 0.58 ≅ 7.5) out of the 13 pots.
Player Hand % win-rate
You (pair) Q♣-6♥-Q♦-4♠-9♦ 42
John (two pairs) 8♣-5♠-8♦-3♥-3♣ 58
Therefore, John will win about 15% of the pots (11:2 against), when only his hand receives improving cards by the fifth street.
In limit games the mathematical expectations of the smaller pair are not good enough despite the larger pot odds the caller is getting. Many players think that going as far as the fifth street is justified because it will cost them two minimum bets only. However, a simple mathematical analysis of the situation suggests that John’s adventure will cost him even when you decide to gamble against his trip Eights. Let us look at the figures again. In every 100 pots, only John’s hand will improve 21 times by the fifth street. But he will lose at least 6 out of the 21 pots in which he makes either two pairs or trips. Therefore, out of every 100 times he gives you a spin, he will win about 120 minimum bets (15 wins × 8 bets
= 120 bets) and lose more than 194 ((6 losses × 6 bets) + (79 losses × 2 bets) = 194 bets). The figures speak for themselves. John will lose the equivalent of 0.74 of a minimum bet every time he gambles to the fifth street.
As regards pot-limit poker, the above analysis proves the futility of gambling with small pairs in pot-limit Seven-Card Stud. The mathematical
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expectations of the small pair are very poor. The best way to play a small pair with a bad kicker against a higher pair is when your pair is in the hole.
Now if your opponent has 25 times as many chips as the value of the bet on third street you must go for it, especially if you know that your opponent will not trash his big pair.
The following hand, which was played in the £100 game at the Victoria Casino in London, will illustrate the potential of hidden pairs against bad players; the game was a 9-handed one in which every player anted £1.
Mike, who is a loose player, had £1,500 in front of him. He was dealt A♣-J♦-A♥. Having the high card, he started the pot by betting £10.
Kevin, who is a good but moderately tight player, called the bet with K ♥-K♠-6♣; he had £1,000 worth of chips stacked in front of him. The dealer dealt the 7♠ to Mike and the K♦ to Kevin. Mike bet £30 and Kevin called and raised £90.The raise was called by Mike! The dealer then gave the J♠ to Mike and the T♣ to Kevin. Now Mike had two pairs Aces and Jacks. He was living in a blessed state of oblivion; he did not realise that he was 9:2 against if Kevin had a set of Kings. Mike bet £270 and Kevin went all-in. They both hit blanks on the next two cards and Kevin celebrated a very nice pot.
Mike made two crucial mistakes in this pot. First, he did not worry about Kevin’s hand. All he could think of was his two Aces. I bet he said to himself, ‘I have got two Aces glued to my palms and I’m going to close my eyes and allow my hands to do the talking.’ This is what I call a state of ‘OBLIVION’. Second, he should have realised that Kevin knew he was challenging two Aces. There was no way Kevin would re-raise in the fourth street, thereby placing all of his £1,000 at jeopardy, unless he could beat Aces-up. On the other hand, Kevin knew that if he struck gold on the fourth street, Mike would not trash his big pair.
The moral of the story is: if you have a small pair in the hole, call for one card only. If you don’t make trips by the fourth street, release your hand, because you do not have the appropriate odds unless you and your opponent have money worth at least 30 times the next bet.
Do not get involved in a raised pot if one of your pair cards is among the discards. However, I would buy one card to see the fourth street if the pot is not raised. When I capture the last card of my pair, I am going to get
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more than 40:1 for my money because many players, who are unfortunate enough to pair their high hole cards, will give me the needed action. On one occasion I was last one to act when the dealer gave me 8-7-8; one of the remaining Eights was discarded by another player. My fourth card was the last Eight in the deck while another player paired the Ace he had in the hole. To cut a long story short, I was handsomely rewarded! At the showdown my opponent said, ‘I did not put you on trips because I thought you were aware of the discards.’ He did not realise that it cost me only the bring-in bet to call on third street, but because one of the Eights was discarded, I am likely to get more than 40 times that amount when my door card is paired.
I hope that by now you have understood the following very important concept in pot-limit. Most of the time you are going for the implied odds and, therefore, the absolute size of the bet you call is not the important factor. What is important is the relative size of the bet in relation to (1) the money you and your opponent have and (2) the probability of capturing the winning card in the next round of betting.