Must pick any topic from unit 2
- classical probability
-axiomatic probability
-frequentism
-bayesianism
they must be different topics.
assessment is to develop two new exam problems for the material covered in this unit.
For each problem, you must provide three things:
The problem itself.
A detailed breakdown of how to solve the problem (written for an audience of fellow students).
An analysis of why it is a good exam problem.
In total, you will be evaluated according to the three criteria below (listed along with questions to consider when thinking about whether you are doing a good job for each). See the assignment rubric for a points breakdown.
Clarity
Would it be possible for a student who encounters the problem to be confused?
How well can another student follow the solution?
Is everything grammatical and well organized?
Breadth/Depth
How much of the material in the chapter is being tested by the problem?
Do students need to understand the tested material, or do the just need to memorize a bunch of stuff?
Are you striking a good balance between breadth and depth?
Thoroughness
How useful might another student find the provided explanation?
Are you demonstrating an understanding of how the problem relates to course material (both what is covered by the problem and what isn't)?
The problems should be different both in their details and in their general structure—e.g., if this was for logic, and one of them gives a natural language sentence to translate, then the other one should not just be a different sentence to translate.
The Example
(this is what i am looking for)
The Problem
Consider the following atomic sentences.
M := "Boris takes medication."
R := "Boris eats rice."
P := "Boris eats pineapple."
S := "Boris is sick."
Given those atomic sentences, which of the following options is the best translation of the natural English sentence "Boris will take medication if and only if he’s sick or about to eat some pineapple.".
a. (M↔︎S)∨(M↔︎P)
b. M↔︎(S∨P)
c. M→(S∨P)
d. (S∨P)→M
Solution
Begin as usual (see Section 3.3 in the notes) by identifying the operator words being used in the sentence.
In this case, you just have
Boris will take medication if and only if he's sick or about to eat some pineapple.
Checking to make sure that you are doing this correctly is made a bit easier compared to doing a translation from scratch since you are given the necessary atomic sentences. The ``Boris will take medication'' part of the sentence is clearly M. Where it says ``he's sick'', it is referring to Boris, so that's S. Lastly, ``about to eat some pineapple'', is talking about Boris eating pineapple, which is P—and, remember, timing doesn't matter, so saying that Boris is about to eat some pineapple is treated the same in the logic as saying that he is (present tense) eating pineapple.
Since we've confirmed that there are atomic sentences for all of the sentence parts other than the identified operator words, we should be doing okay so far.
Next you need to identify the operators that go with the operator words. In this case, you just need to remember that ``if and only if'' is the biconditional ↔︎, and that ``or'' is the disjunction ∨. Having done that, we can immediately rule out options (c) and (d), since they feature material conditionals instead of biconditionals.
We are left, then, with two options:
a. (M↔︎S)∨(M↔︎P)
b. M↔︎(S∨P)
You could just read off both of these in logic-ese and see that (b) is the correct answer. Option (a) reads as
Boris takes medication if and only if Boris is sick, or Boris takes medication if and only if Boris eats pineapple.
while (b) reads as
Boris takes medication if and only if Boris is sick or Boris eats pineapple.
which is almost the original sentence verbatim.
Alternatively (and more reliably) you should think about what these sentences are actually saying—i.e., when they are true or false—and compare them to what is being said by the original sentence. Specifically, you should identify a possible world where the two options end up with different truth values and then see which one has the value that agrees with the original sentence.
In this case, you could consider a world in which Boris is sick, but doesn't eat pineapple and doesn't take medication. Substituting those truth values into option (a)—i.e., replacing the Ms and Ps with 0 and the Ss with 1—gives us (0↔︎1)∨(0↔︎0). Since the biconditional is true if and only if its operands have the same truth value, then we can simplify that down to 0∨1. Then, since the disjunction is true if and only if at least one of its operands is true, we can see that (a) is true in this world.
In contrast, (b) ends up false: Substitution gives us 0↔︎(1∨0), which (according to the just noted semantics for the disjunction) simplifies to 0↔︎1. And 0↔︎1 (according to the just noted semantics for the biconditional) is false.
Our two options are different in this world—(a) is true, while (b) is false—so now we just have to think about what the truth value you would be for the original sentence. It is, again,
Boris will take medication if and only if he's sick or about to eat some pineapple.
According to this, being sick is sufficient for Boris to take medication. So, if he is sick---and he is in the possible world we are considering---then he should be taking medication. But he isn't taking medication in our considered possible world. So the original sentence would be false in that world, and so to should a proper translation of it be false in that world. That means that (a) can't be the correct translation since it comes out true, leaving (b) as the correct answer to the problem (by process of elimination).
Analysis
This is a good exam problem for testing a student's understanding of translations, the biconditional, disjunction, and, to some extent, the material conditional. As is apparent from the provided solution, there are two major realizations that need to made in order to arrive at the solution: (1) that "if and only if" is the biconditional and not the material conditional, and (2) that the biconditional, and not the disjunction, is the major operator of the sentence.
Notably, the problem is very targeted in its testing of those two things. Because it provides the atomic sentences, trying to figure out what the atomic sentences should be isn't a potential confound in identifying the solution. However, this targeting goes beyond simply asking the student to regurgitate the respective definitions of disjunction and the material and bi- conditionals. Arriving at the correct answer depends on knowing those definitions, but it also depends on understanding them and how they can be applied to understand the meaning of a complex natural English sentence.
Rubric
Exam Problems Rubric
Exam Problems Rubric
Criteria Ratings Pts
This criterion is linked to a Learning Outcome Problem 1: Clarity 5.0 pts
This criterion is linked to a Learning Outcome Problem 1: Breadth/Depth 10.0 pts
This criterion is linked to a Learning Outcome Solution 1: Clarity 10.0 pts
This criterion is linked to a Learning Outcome Solution 1: Thoroughness 10.0 pts
This criterion is linked to a Learning Outcome Solution 1: Breadth/Depth 5.0 pts
This criterion is linked to a Learning Outcome Analysis 1: Clarity 5.0 pts
This criterion is linked to a Learning Outcome Analysis 1: Thoroughness 5.0 pts
This criterion is linked to a Learning Outcome Problem 2: Clarity 5.0 pts
This criterion is linked to a Learning Outcome Problem 2: Breadth/Depth 10.0 pts
This criterion is linked to a Learning Outcome Solution 2: Clarity 10.0 pts
This criterion is linked to a Learning Outcome Solution 2: Thoroughness 10.0 pts
This criterion is linked to a Learning Outcome Solution 2: Breadth/Depth 5.0 pts
This criterion is linked to a Learning Outcome Analysis 2: Clarity 5.0 pts
This criterion is linked to a Learning Outcome Analysis 2: Thoroughness 5.0 pts
Total Points: 100.0
Criteria Ratings Pts
This criterion is linked to a Learning Outcome Problem 1: Clarity 5.0 pts
This criterion is linked to a Learning Outcome Problem 1: Breadth/Depth 10.0 pts
This criterion is linked to a Learning Outcome Solution 1: Clarity 10.0 pts
This criterion is linked to a Learning Outcome Solution 1: Thoroughness 10.0 pts
This criterion is linked to a Learning Outcome Solution 1: Breadth/Depth 5.0 pts
This criterion is linked to a Learning Outcome Analysis 1: Clarity 5.0 pts
This criterion is linked to a Learning Outcome Analysis 1: Thoroughness 5.0 pts
This criterion is linked to a Learning Outcome Problem 2: Clarity 5.0 pts
This criterion is linked to a Learning Outcome Problem 2: Breadth/Depth 10.0 pts
This criterion is linked to a Learning Outcome Solution 2: Clarity 10.0 pts
This criterion is linked to a Learning Outcome Solution 2: Thoroughness 10.0 pts
This criterion is linked to a Learning Outcome Solution 2: Breadth/Depth 5.0 pts
This criterion is linked to a Learning Outcome Analysis 2: Clarity 5.0 pts
This criterion is linked to a Learning Outcome Analysis 2: Thoroughness 5.0 pts
