Section A.2 Probability Theory Review
This is an extremely brief “review” of the limited probability theory we utilize in
Chapter 5. It’s not even particularly fair to call this a review, since probability is not a prerequisite to this course. However, the limited amount we use is fairly straightforward and intuitive.
If a more thorough treatment is needed, then depending on your goals, there are good options available. For someone looking to explore some elementary probability theory, the introductory statistics textbook
“OpenIntro Statistics” by David Diez, Christopher Barr, and Mine Çetinkaya-Rundel does a good job presenting this material. It also is an excellent introducotry statistics text with labs and data available. For a calculus-based, theory heavy treatment of this text, I recommend
“Probability: Lecture and Labs”.
Definition A.2.1.
In probability, an experiment is an occurrence with a measurable result. Each instance of an experiment is a trial. The possible results of each trial are called outcomes. The set of all possible outcomes for an experiment is the sample space for that experiment.
Definition A.2.2.
Given an experiment with sample space \(S\text{:}\)
Definition A.2.4.
A random variable is a function from sample space to an outcome set. For our purposes, this set of outcomes will always be \(\mathbb{R}\text{.}\)
A probability distribution is, roughly speaking, a complete description of a random variable and the likelihood of each output. In the case of random variables with a finite number of possible outputs a probability distribution table is a convenient way of presenting this information.
Example A.2.6. Poisoned apples.
Snow White has a basket of 10 apples, 3 are poisoned. She is going to pick 4 apples at random to eat for some reason. Let \(X\) denote the number of poisoned apples she eats.
The probability distribution for \(X\) would be:
\begin{equation*}
\begin{array}{|c|cccc|}
\hline
x \amp 0 \amp 1 \amp 2 \amp 3 \\
\hline
P(X=x) \amp \frac{{3\choose 0}{7\choose 4}}{ {10\choose 4} } \amp \frac{{3\choose 1}{7\choose 3}}{ {10\choose 4} } \amp \frac{{3\choose 2}{7\choose 2}}{ {10\choose 4} } \amp\frac{{3\choose 3}{7\choose 1}}{ {10\choose 4} }\\
\hline
\end{array}
\end{equation*}
equivalently:
\begin{equation*}
\begin{array}{|c|cccc|}
\hline
x \amp 0 \amp 1 \amp 2 \amp 3 \\
\hline
P(X=x) \amp \frac{35}{210}\amp \frac{105}{210} \amp \frac{63}{210} \amp \frac{7}{210}\\
\hline
\end{array}
\end{equation*}
or:
\begin{equation*}
\begin{array}{|c|cccc|}
\hline
x \amp 0 \amp 1 \amp 2 \amp 3 \\
\hline
P(X=x) \amp \approx 0.1667\amp 0.5 \amp 0.3 \amp \approx 0.0333\\
\hline
\end{array}
\end{equation*}
This can be seen by the following R simulation:
Definition A.2.7.
Given a finite random variable \(X\text{,}\) it’s expected value is the predicted average outcome of experiments, and is computed:
\begin{equation*}
E(X)=\sum P(X=x)\cdot x.
\end{equation*}
Note that the “Expected Value” may not be a value we actually expect, that is, may not be one of the outcomes, just an average outcome. We think of this as the outcomes of \(X\text{,}\) weighted by their likelihood, so the more likely outcomes contribute more than the less likely ones.
Example A.2.8.
Recall
Example A.2.6. The expected value of poisoned apples would be
\begin{equation*}
E(X)=0\cdot\frac{35}{210}+1\cdot\frac{105}{210}+2\cdot\frac{63}{210}+3\cdot\frac{7}{210}=1.2.
\end{equation*}
We can compute the mean of the previously simulated number of poisoned apples and visualize it:
