Date archives "January 2017"

First Impressions of STAT 302 – Probability

I’m going to start this entry off by saying this course is incredibly interesting, but is by far one of the hardest classes I have ever had to take.

The class started off simple. It felt like review of MATH 220 to me: the union, intersection, and complements of probabilities acted similar to those of sets. For example:
Sets: Let the set A={1,2,3} and B={3,4,5}

A∪B= A+B-(A∩B) = {1,2,3,4,5}

Probability: Let the probability of event A=1/3 and B=1/2

A∪B= A+B-(A∩B) = 1/3 + 1/2 – 1/3 = 1/2

The next section was Combinatorics: Counting, Permutations and Combinations. I remember learning about this in grade 12, but did not go in depth. The questions we were expected to be able to do in this course were extremely complicated and I still believe that this section is one of the most difficult ones in the whole textbook. It forces us to think critically and even creatively, as these questions usually have more than one way of solving it.

One of the questions on the FIRST assignment: A quiz consists of 10 true/false questions. A student decides that he will not answer FALSE for any two consecutive questions. In how many ways can he answer all 10 questions?

The question seems quite simple to begin with. As soon as I tried to solve it, it was as if the question’s difficulty was increasing at an exponential rate. A random classmate of mine and I discussed our strategy in solving it and his solution looked like:

So for this question, drawing out all the combinations is possible but not very efficient. There was also talk among other classmates that it followed a Fibonacci sequence. My thought process was that there must be less than 6 false answers in order for none to be consecutive, then using the nCr (n choose r) formula. But the solution is much more complicated than that, which I will not go into on my blog post.

 

An Introduction to MATH 220 – Logic and Statements

Just finished set theory a few weeks ago in Math 220 and I honestly found it quite interesting. The union, intersection, partition, and complement of sets all relate to probability in Stat 302. One may say that the intersection of the content of the 2 courses is non-empty… Jokes aside, props to Professor Brett Kolesnik for having engaging lectures taught at the right pace. You know a course is well taught when you rarely need to look at textbook explanations.

Anyways, after set theory came logic and statements. This came quite naturally to me due to my prior programming experience in first and second year. Simple operators such as:

Not/Negation (~)

Or/Disjunction (∨)

And/Conjunction (∧)

If/Implication (⇒)

If and only if/Biconditional (⟺)

had truth tables that I already knew, simply by thinking in terms of code. However, something new came up that troubled me. Logical equivalence theorems such as commutative, associative, distributive, and De Morgan’s laws were not anything I was familiar with. After some practice it was much easier to handle.

Simple example: Show that ~(P⇒Q) is logically equivalent to (P∧~Q)∨(Q∧~P)

Start with ~(P⇒Q)≡(P∧~Q)∨(Q∧~P)

~(P⇒Q)≡~((P⇒Q)∧(Q⇒P))      … by definition of biconditional

≡~(P⇒Q)∨~(Q⇒P)     … by De Morgan’s Law

≡(P∧~Q)∨(Q∧~P)       …∎