[This is the course I'm working on right now! - Kevin]
In the tutorial course on different interpretations of the probability concept I often referred to the “probability calculus”, or the “mathematical theory of probability”. This is what most people are talking about when they refer to “probability theory”. Probability theory provides basic rules for reasoning with probabilities. Given probabilities for events A and B, we can calculate the probability of “A and B”, “A or B”, “A given B”, and so on.
These rules are part of our modern understanding of how we ought to reason about uncertainty. In this respect they function in a way similar to the rules of deductive propositional logic, which tell us how we ought to reason with truth-preserving inferences that involve conjunctions, disjunctions, conditionals, etc.
Critical thinking literacy requires that we be at least somewhat familiar with probabilistic relationships (like, we should all know that the probability of two events both occurring is always smaller than the probability of either event occuring separately). More importantly, the fallacies of probabilistic reasoning that we’ll be looking at in the next course can only be understood in relation to these basic rules.
Working through simple problems is a fun way to learn the basic concepts (and some concepts can only be grasped by working through examples) but critical thinking doesn’t require that we become probability theory masters!
Introduction
PART I: PRELIMINARY CONCEPTS
1. What has a probability? Propositions versus Events (3:29) (FREE)
2. Probabilities range between 0 and 1 (4:17)
3. Mutually exclusive events (0:58)
4. Independent events (3:19) (FREE)
PART II: THE BASIC RULES
1. The Negation Rule: P(not-A) (5:18)
The Disjunction Rules: P(A or B)
2. Restricted Disjunction Rule: P(A or B) = P(A) + P(B) [used when A and B are mutually exclusive] (4:39)
3. General Disjunction Rule: P(A or B) = P(A) + P(B) – P(A and B) (5:17) (FREE)
The Conjunction Rules: P(A and B)
4. Restricted Conjunction Rule: P(A and B) = P(A) x P(B) [used when A and B are independent] (3:26)
5. General Conjunction Rule: P(A and B) = P(A) x P(B/A) (5:13)
Conditional Probability Rules: P(A given B)
6. General Conditional Probability Rule (8:38) (FREE)
7. Total Probability Rule (6:50)
8. Bayes’ Rule (7:19)
PART III: WORKED EXAMPLES
You’re encouraged to try these on your own.
Click here to download the problem set below in pdf format. (Members can download the worked solutions.)
1. In a box there are 3 red pens, 5 blue pens, and 2 black pens. If a person selects a pen at random, what is the probability that the pen is
a. a blue or a red pen?
b. a red or a black pen?
2. At a political rally there are 8 Democrats and 10 Republicans. 6 of the Democrats are females and 5 of the Republicans are females. If a person is selected at random, what is the probability that the person is either a female or a Democrat?
3. What is the probability of tossing a coin ten times and getting ten heads in a row?
4. What is the probability of getting this particular sequence – H, H, T, H, T, T, H, T, H, T – in ten coin tosses?
5. What is the probability of drawing two aces from a regular deck of cards in two draws:
a) if the first card is replaced before the second is drawn?
b) if the first card is NOT replaced before the second is drawn?
6. This is a typical conditional probability problem. Imagine two urns, each containing red and green balls. Urn A has 80% red balls and 20% green balls. Urn B has 40% red and 60% green. You pick an urn at random by tossing a coin, and then draw a ball from the urn you chose (remember, you don’t know which urn you’ve drawn from). The aim is to guess which urn you’ve drawn from, based on the color of the ball.
You draw a red ball. What is the probability that you’ve drawn the ball from urn A?
7. Taxi Hit and Run (a classic application of Bayes’ Rule, and a good illustration of the “base rate fallacy”):
A town has two taxi companies, Blue Cab and Yellow Cab, named after the color the taxis are painted. 95% of the taxis in town are owned by Yellow Cab; Blue Cab owns the rest.
A taxi dents a parked car on the street and speeds away. A witness reports to the police that the taxi was blue.
Psychological studies have shown that witnesses to a single-car accident are 80% likely to be able to report the color of the car correctly.
Question: Given the eye witness report, and assuming that this witness is typical, how likely is it that a blue taxi was really the culprit?
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