Mental Model Theory—Illusion of Possibility Example
(I have written an overview of the mental model theory which is in main and the link is here. You should read this overview before you read this post. You should only read this post if you want more explicit details on the first example which demonstrates the illusion of possibility)
Consider the following problem:
Before you stands a card-dealing robot. This robot has been programmed to deal one hand of cards. You are going to make a bet with another person on whether the dealt hand will contain an ace or whether it will contain a king. If the dealt hand is just a single queen, it’s a draw. Based on what you know about this robot, you deduce correctly that only one of the following statements is true.
The dealt hand will contain either a king or an ace (or both).
The dealt hand will contain either a queen or an ace (or both).
Based on your deductions, should you bet that the dealt hand will contain an Ace or that it will contain a King?
If you think that the ace is the better bet, then you would have made a losing bet. In short, this is because it is impossible for an ace to be in the dealt hand.
To see why this is I will list out all of the explicit mental models.
Below are the mental models that people will create in accordance with the principle of truth. (See the article in main for what this is). You can see that Ace is in both rows, which makes it seem like ace must obviously be more likely to be in the dealt hand.
|
Statement 1 true |
K |
A |
K ∩ A |
|
Statement 2 true |
Q |
A |
Q ∩ A |
But, when we look at the full explicit set of potential models (including the models when one of the statements is false) we will realise that it is impossible for an ace to be in the hand. Note that ¬ stands for negation. (¬A) means that the hand does not have an ace. The first possible scenario is when statement one is true and statement two is false. The mental models for this are in the below table:
|
Statement 1 true Statement 2 false |
K |
A |
K ∩ A |
|
¬Q |
¬A |
¬Q ∩ ¬A |
Consider each column after the first as a potential possibility for how the dealt hand could be.
The first column means that the dealt hand will have a king and not have a queen. This looks good. There are no problems with this.
The second column means that the dealt hand will have an ace and not have an ace. We have reached a contradiction, which implies that this possibility is impossible.
The third column is also impossible as the first row has (A) and the second has (¬A).
If we look at the second possible scenario which is when statement two is true and statement one is false, then we get the below table.
|
Statement 2 true Statement 1 false |
Q |
A |
Q ∩ A |
|
¬K |
¬A |
¬K ∩ ¬A |
Once again if we can consider each column after the first as a potential possibility for how the dealt hand could be.
The first column means that the dealt hand will have a queen and not have a king. This looks good. There are no problems with this.
The second column like in the first table is a contradiction and so is impossible.
The third column is also a contradiction and so is impossible.
If we remove the ace possibilities as this leads to contradictions, we end up with the below table:
|
Statement 1 true Statement 2 false |
K |
|
¬Q |
|
|
Statement 2 true Statement 1 false |
Q |
|
¬K |
This table has two possibilities. The dealt hand contains a king or the dealt hand contains a queen. Knowing this, we can know say that it is more likely for there to be a king in the dealt hand as it impossible for an ace to be in the hand. Therefore, we should bet that there is a king in the hand.
So, we have
We don’t have both “Either K or A” and “Either Q or A”
Therefore, we either have “Neither K nor A” or “Neither Q nor A”
Since both of the possibilities involve “no A”, there can be no A.
Your post seems to be a rather verbose way of showing something that can be shown in three lines. I guess you’re trying to illustrate some larger framework, but it’s rather unclear what it is or how it adds anything to the analysis, and you haven’t given the reader much reason to look into it further.
The reason that someone might think an Ace would be a good choice is that they misread it as saying “one of these two statements is true”. But it is nowhere stated that either statement is true; rather it is stated that at least one statement is false. Once one notices that the Ace is involved in both of these statements, of which one has to be false, one’s intuition should lead one choosing the King.
Also, if you’re using set notation, (K ∪ A) indicates the same thing as (A or K or K ∩ A).
I have rewritten the header to this post to make it clear that you should read the post in main first and only look at this one if it is required.
Technically the problem is very simple, but it does frequently fool people. If you write out the logic of it like in the above post, then people will very easily get the right answer. This post is meant to be a verbose explanation of the solution for people who don’t believe that you should choose the king. You can read this post if you want to know why people get fooled by this simple problem.
This is the example as it is written in the academic literature
Only one statement about a hand of cards is true:
There is a King or Ace or both.
There is a Queen or Ace or both.
Which is more likely, King or Ace?
Most people say Ace.