OCW 6.041SC
Lecture 1 Probability Models & Axioms
Lecture 2: Conditioning and Bayes’ Rule
Textbook – Introduction to Probability – Bertsekas & Tsitsiklis
1.1 Sets
1.2 Probabilistic Models
1.3 Conditional Probability
1.4 Total Probability Theory & Bayes’ Rule
Looking forward at the plans I’m formulating for future studies, I’ve daydreamed about what math courses might come next. Yes, I said ‘daydream’. Now that I’ve completed my Calc and Linear Algebra, I feel like I’ve finally completed my initiation, built my foundation and can explore more freely.
I ultimately chose MIT OCW 6.041 Probabilistic Systems Analysis and Applied Probability with Professor John Tsitsiklis. Probability really seemed like the linchpin, or at least greatest ROI course of study, for future Economics and Data pathways. It’s basically a requirement for the DEDP, which is the primary goal. I grow increasingly interested in Operations Research and hard, applied mathematics. There again, Probability seems like the next logical step.
This course is great. It’s replete with all the lectures, recitations, problem sets, quizzes, etc. And Professor Tsitsiklis is great. He describes things well, is organized in his lectures, and is personable. Another big win – OCW is amazing.
I’ve read that this is a challenging course as probability is counterintuitive to our primitive ape brains. It feels different than Calc. More in a scratch my chin, sink into philosophical logic-type thought difficult. I like it. Granted, I’ve just gotten started.
It starts out wonderfully, with the few simple rules/frameworks about sample spaces, events, and probabilities, and the three axioms upon which the forthcoming method of thought will be built. Namely, 1) Non-negativity, 2) Normalization, and 3) Additivity. It comes along with some new notation, but nothing that takes too long to figure out. Questions can elicit knee-jerk answers (almost a sure-fire way to know I’m wrong), exposing my shortcomings in evaluation probability.
Examples are excellent, as they relate to real-world situations, some we all associate with probability, such as cards, coins, and dice. But then there are others, like the radar detection scenario returned to several times, or the classic Monty Hall problem. What I found particularly interesting was an example called False Positive Puzzle. It goes something like this:
A test for a rare disease is assumed to return correct results 95% of the time: if a person has the disease, the test results are positive with a probability of 0.95, and negative with the same probability if the person doesn’t have the disease. A random person has a probability of 0..001 of having the disease. Given that a person tested positive, what’s the probability that they have the disease? Now it was reported by The Economist that 80% of those questioned at leading hospitals got this type of question wrong; most thought the answer was 95%. The math?
\begin{aligned} A &= \text{has the disease}, \qquad B = \text{tests positive} \\[14pt] P(A\mid B) &= \frac{P(A)P(B\mid A)}{P(A)P(B\mid A)+P(A^c)P(B\mid A^c)} \\[16pt] &= \frac{(0.001)(0.95)}{(0.001)(0.95)+(0.999)(0.005)} \\[16pt] &\approx 0.1598 \end{aligned}
How cool is that?!
OK, so I am very excited to be taking this course. I want more. Feed me, Seymour!
Until later…