Week 1

This is an outline of the topics we covered in the first week of class. These notes are not a substitute for your own note-taking. I highly recommend that you take your own notes during class. If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes.

Tuesday 1/13

In calculus, you mostly asked deterministic questions about functions. In science experiments, you need to take randomness into account.

A toxin molecule in a cell has a certain chance each minute to leave the cell.

A patient takes a diagnostic test for a disease and wants to know the chance that they have the disease based on the test result.

Based on historical year-over-year population growth rate data for a particular species, what is a reasonable range for next year's population?

The sample space, often denoted \Omega, is the set of all possible results of an experiment. A single result is called an outcome, while a collection of results is called an event.

Let A be a set. The symbol \in means "is an element of", as in a \in A. Given another set B, we say A is a subset of B, written A\subset B, to mean that every element of the set A is also an element of the set B.

An experiment consists of rolling a standard 6-sided die. The sample space is \Omega = \{1, 2, 3, 4, 5, 6\}. One possible event is A = \{2, 4, 6\}, i.e., the event that the result of the roll is even.

Consider sets A and B, each contained inside \Omega. We can combine sets in a variety of ways:

  • Union

    The union of A and B is the set A \cup B = \{x \mid x \in A \text{ or } x \in B\}.

  • Intersection

    The intersection of A and B is the set A \cap B = \{x \mid x \in A \text{ and } x \in B\}.

  • Difference

    The set difference A-B is the set A - B = \{x \mid x \in A \text{ and } x \notin B\}.

  • Complement

    The complement of A is the set A^c = \{x \in \Omega \mid x \notin A\}.

  • Empty Set

    The empty set, usually written \emptyset or \{\}, is the set which contains no elements.

    • It's useful sometimes to draw pictures called Venn diagrams representing sets:

    Example Venn Diagram

    Venn diagram showing sets A, B, C with the region representing (A\cup B\cup C) - (A \cap C) shaded.

    \begin{tikzpicture} \def\firstcircle{(90:1.75cm) circle (2.5cm)} \def\secondcircle{(210:1.75cm) circle (2.5cm)} \def\thirdcircle{(330:1.75cm) circle (2.5cm)} \fill[gray!30] \firstcircle; \fill[gray!30] \secondcircle; \fill[gray!30] \thirdcircle; \begin{scope} \clip \firstcircle; \clip \thirdcircle; \fill[white] \firstcircle; \end{scope} \draw \firstcircle node[text=black,above] {$A$}; \draw \secondcircle node [text=black,below left] {$B$}; \draw \thirdcircle node [text=black,below right] {$C$}; \node at (0, -4.5) {$(A\cup B\cup C) - (A \cap C)$}; \end{tikzpicture}

    Next, we want to start assigning probabilities to each individual outcome so we can then find the probabilities of events.

    An experiment consists of rolling a standard 6-sided die. The sample space is \Omega = \{1, 2, 3, 4, 5, 6\}. We might assign probabilities as follows:

    Distribution for a fair die x \Pr(x) 1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/6

    Note that we don't have to assign the same probability to each outcome. If we do, we call this distribution uniform. If we have some event, such as A = \{2, 4, 6\}, then we calculate the probability of the event by adding together the probabilities of the outcomes that make up the event: \Pr(A) = \Pr(2) + \Pr(4) + \Pr(6) = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} Note that this is the same result that we would get if we counted the total number of outcomes in A and divided by the total number of outcomes in \Omega.

    Given a finite set A, the cardinality of A, written |A|, is the number of elements in A.

    If \Omega is a finite probability space with the uniform distribution and A \subset \Omega is an event, then: \Pr(A) = \frac{|A|}{|\Omega|}

    Suppose we have a weighted die that's much more likely to come up 6 than any other outcome.

    Distribution for a fair die x \Pr(x) 1 0.1 2 0.1 3 0.1 4 0.2 5 0.1 6 0.4

    With this distribution, if A = \{2, 4, 6\}, then: \Pr(A) = 0.1 + 0.2 + 0.4 = 0.7 \neq \frac{3}{6}.