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Drill tip: Funky “most” relationships 1
BinghamtonDave
Alum Member ⭐ 🍌🍌
Hey folks, I wanted to post a few questions here that contain “most” relationships in them that are not that common. They collectively might only take 45 minutes to memorize, and who knows: one might show up on your exam and you can get the point! This is part 1 in what I hope is a 3 part series examining weird most relationships.
The first most relationship I want to discuss takes this following form:
Most As are Bs
Most Bs are not As
The following are 2 questions that make use of this relationship, we are asked for a must be true and the answer to both is the same iteration of the must be true.
Pt 77-4-13
Pt 80-1-22
I have left one other question out. As an exercise, I want you, the reader to try to memorize this form and see if you come across it yourself, that way you can see if you have actually retained the memory and understanding of the form and if you would net the point come test day.
Lets take a look at what these two statements mean. Mr. Ping points this process out in the lessons, but it is worth repeating.
Here is an example:
Most people who live in the United States know who The Beatles are
But, most people who know who The Beatles are do not live in the United States
In combination, what must be true here?
Let’s pretend there are 320 people in the United States and lets assign a number constituting “most” of that set to be those who know who The Beatles are: 200. It might be helpful here to view “most” as simply something equal to or more than 51%. So that assignment of 200 is our “most.” This is a very basic way to look at this concept, there might be more thorough ways to look at it, but I think this will work.
Now let’s rephrase that first statement with something more concrete:
200 out of a total of 320 people in the USA know who the Beatles are
Now let’s look at the other statement and its relationship with the first statement:
But, most people who know who The Beatles are do not live in the United States
Here we are looking at the total set of people who know who The Beatles are and we are making a statement about that set: that that 200 we know comes from the United States constitutes less than half, or less than what we are colloquially calling “most” or more specifically, a maximum 49%, this has to be true in order to make room for the number of people that constitute 51% or more not from the USA who know who The Beatles are. So whatever the rest of that set is (not USA): it must be something more than 200, because if it were less, the statement itself would be false. So what raw number constitutes most in one context does not constitute most in another context.
So the second statement tells us that the 200 we have from the USA falls short of most, and the remainder of the total set of people who know who The Beatles are comes from Not USA. So the total set of people who know who The Beatles are might be 1,000 and 200 come from the USA and 800 from elsewhere/not USA. That is a fact pattern commensurate with our statements. A number that constitutes most (200) in one context is less than half in another context.
Most professional Bull riders are from Texas
Most people from Texas are not professional bull riders
This is an example that might make the idea of sets in this context easier to grasp.
Texas has roughly 30 million people, a set much larger than “professional bull riders.”
In one context “most” means something like 180/300 (bull riders)
And in another context “most” would mean something like 16 million (From Texas)
Intuitively, this makes sense, and I submit so does the must be true inference: that there are more people from Texas than there are professional bull riders.
I have designed a small exercise to help demonstrate this point. Below are 3 total sets that reference our Beatles/USA example above, which one is not commensurate with our fact pattern in the stimulus? Here is a reminder of what the fact pattern in the stimulus is:
Most people who live in the United States know who The Beatles are
But, most people who know who The Beatles are do not live in the United States
1.
Total set:700 know who the Beatles are
USA:200/320 know who The Beatles are
Not USA:500 out of a total set of 700 know who the Beatles are
2.
Total set:300 know who the Beatles are
USA:200/320 know who the Beatles are
Not USA:100 out of a total set of 300 know who the Beatles are
3.
Total set:200,000 know who the Beatles are
USA:200/320 know who The Beatles are
Not USA:199,800 out of a total set of 200,000 know who The Beatles are
That’s correct, set (2) does not fit our fact pattern.
Now here is another intuitive example:
Most people who currently attend Harvard Law School are really smart
Yet, most people in the world who are really smart, don’t currently attend Harvard Law school
Intuitively, we know that the second set “really smart” is a much large set than the total group currently attending Harvard Law school.
Now, try applying a fact pattern to the two questions above and see what emerges as a must be true! Keep this form in mind, it might be rare, but getting it correct could be the difference between a 159 and a 160, or getting it correct quickly and efficiently could free up time that could lead to getting 3 questions correct you otherwise might have been pressed for time when you got to.
You know the drill: I'm open to correcting any mistakes I might have made here.
Thank you for reading!
David
Comments
I fully support a motion that would make this a weekly series of tips from Dave.
I second!
I third!!!
Motion carried!
Thank you. This was very informative and insightful.
I was about to ask if there was a way to get notifications whenever Dave posts a thing. TYVM.
I was about to ask if there was a way to get notifications whenever Dave posts a thing. TYVM.