Definitions¶
We start with some basic definitions:
Definition: sample space¶
A sample space is the set of all possible outcomes of an experiment.
Definition: event¶
An event is a subset of the sample space.
Definition: naïve definition of probability¶
Under the naïve definition of probability, the probability of a given event $A$ occurring is expressed as
\begin{align} P(A) &= \frac{ \text{# favorable outcomes}}{\text{# possible outcomes}} \end{align}
assuming all outcomes are equally likely in a finite sample space.
Counting¶
With the multiplication rule, if we have an experiment with $n_1$ possible outcomes; and we have a 2nd experiment with $n_2$ possible outcomes; ..., and for the rth experiment there are $n_r$ possible outcomes; then overall there are $n_1 n_2 ... n_r$ possible outcomes (product).
Let's say you are ordering ice cream. You can either get a cone or a cup, and the ice cream comes in three flavors. The order of choice here does not matter, and the total number of choices is $2 \times 3 = 3 \times 2 = 6$. This can be represented with a very simple probability tree.
The binomial coefficient is defined as
\begin{align} \binom{n}{k} = \begin{cases} \frac{n!}{(n-k)!k!} & \quad \text{if } 0 \le k \le n \\ 0 & \quad \text{if } k \gt n \end{cases} \end{align}This expresses the number of ways you could choose a subset of size $k$ from $n$ items, where order doesn't matter.
Sampling¶
Choose $k$ objects out of $n$
| ordered | unordered | |
|---|---|---|
| w/ replacement | $n^k$ | ??? |
| w/o replacement | $n(n-1)(n-2) \ldots (n-k+1)$ | $\binom{n}{k}$ |
- ordered, w/ replacement: there are $n$ choices for each $k$, so this follows from the multiplication rule.
- ordered, w/out replacement: there are $n$ choices for the 1st position; $n-1$ for the 2nd; $n-2$ for the 3rd; and $n-k+1$ for the $k$th.
- unordered, w/ replacement: ???
- unordered, w/out replacement: the binomial coefficient; think of choosing a hand from a deck of cards.
Out of the 4 ways of choosing $k$ objects out of $n$, the case of unordered, with replacement is perhaps not as clear-cut and easy to grasp as the other three. Move on to Lecture 2.
View Lecture 1: Probability and Counting | Statistics 110 on YouTube.