The Defense Attorney's Fallacy

The Defense Attorney's Fallacy is a flaw in statistical reasoning that often arises in legal arguments, and the opposite of the Prosecutor's Fallacy. The Defense Attorney's Fallacy is to focus on the unlikelihood of an event while neglecting other available information that makes the event more likely. Consider the following real-world example, from the O.J. Simpson trial:

Blood found at the scene of the murder, when analyzed in a laboratory, matched that of O.J. The characteristics used to perform the match are shared by around 1 in 400 people. The defense attorney argued that if you gathered all of the people in Los Angeles with matching blood, they could fill an entire football stadium and therefore the evidence was useless. While it's true that the blood match evidence proves little on its own, the defense attorney is wrong that the evidence is useless. The defense attorney's fallacy is to ignore all of the other evidence. While there are a large number of people with matching blood, a vanishingly small number of people had matching blood and knew the victims and had motives for murdering them.

What does this have to do with poker? The Defense Attorney's Fallacy crops up when a passive opponent makes an unexpectedly aggressive move, and you just can't bring yourself to believe that they have a good hand. Let's say you're on the button in $1/$2 Limit Texas Hold'em and dealt:

TsAd

A weak-passive player (the villain) limps in before you, you raise, the blinds fold, and the villain calls. You've been watching this player for some time, and you can summarize their entire strategy as follows:

  • Enter the pot with the top 50% of hands (raise with the top 5%)
  • On later streets, bet with at least two pair
  • Call the flop or turn with at least one pair or a flush or open-ended straight raw.
  • Call the river with at least the second highest pair
  • Never bluff

The flop comes:

Tc2d8h

You have top pair, top kicker and you're up against a calling station. What could be better? The villain checks, you bet, and the villain calls. The turn comes:

As

Now you have top two pair. Again, the villain checks, you bet, and the villain calls. The river comes:

7d

This time the villain comes out betting. Should you raise? What is the probability that the villain actually has a better hand? To beat you, the villain must have either a staight or three of a kind. Assuming the last card completed his hand, the villain would need specifically J9 or 77:

Jh9s7c7s

Out of the 990 possible combinations of cards52 card deck - 7 cards seen = 45 remaining cards.
45 choose 2 = 990 combinations the villain may have, there are 16 possible ways the villain could make J9 and 3 possible ways he could make 77. From this, you can conclude that there is a meager 2% chance that the villain can beat us—which means you should raise. If you do that, you're guilty of the Defense Attorney's Fallacy: ignoring all of the other evidence.

Based on what we know of the opponent, what would they bet with? Since we know this opponent well, we know they would only bet with two pair or better. There are many combinations of cards that might give the opponent two pair, and we beat or tie all of them. However, with any combination other than T7 or 87s, the opponent would have bet out earlier or folded before the flop. Based on our observations of this player, the only hands they could have are the following: T7, J9, 87s, or 77.

Due to cards already in use, there are only 6 ways to make T7, 2 ways to make 87s, 16 ways to make J9, and 3 ways to make 77. That gives the villain a (16+3)/(6+16+3+2) = 70% chance of winning! Your previously formidable hand has become an underdog and you should call.

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