The first lecture focuses on experiments and counting all possible
outcomes. Particular attention is paid to all different ways in which
samples can be taken: ordered/unordered and with/without
replacement. The number of possible such samples of size $n$ from a
"population" (set) of $m$ is given in the following table:
n samples from m
ordered
unordered
w/o replacement
$m^{(n)}=m(m-1)\cdots (m-n+1)$
${m\choose n}=\frac{m!}{n!(m-n)!}$
w/ replacement
$m^n$
${m+n-1\choose n}$
Can you explain why? Ask yourself before and after watching the video.
Find your own concrete examples of experiments where each type of
sampling is most natural or appropriate.
Lecture 2
This lecture concludes the discussion about sampling that began in
Lecture 1 above and discusses the binomial formula.
Make sense of the binomial formula
$$
\bigl( x_1+x_2+\dots +x_n\bigr)^m =\sum _{m_1+\dots+m_n=m}\,{m \choose
m_1,m_2,\dots, m_n}x_1^{m_1}x_2^{m_2}\cdots x_n^{m_n}
$$
in different ways by trying to
derive it on your own without peeking it or retrieving it from
memory. You may start with the simpler case when $n=2$ and generalize
from there.
Lecture 3
Here the central object of study of this class is introduced:
probability. The starting point is the so-called sampling
space, which contains all possible outcomes of a (random)
experiment. A sample space is simply a set $S$ the elements of which
we think of as outcomes and a probability is a real-valued
function defined for subsets (events) $E$ of the set $S$ which satisfies the
following properties:
$P(E)\in [0,1]$ for any $E \subset S$.
$P(S)=1$.
$P \bigl( \bigcup _{j=1}^\infty E_j\bigr)=\sum _{j=1}^\infty
P(E_j)$ whenever $E_j\cap E_k=\emptyset$ for all $j\neq k$.
Two of the three defining properties of a probability are
quite intuitive, the third is more of a technical nature (a sort of
continuity property) and will be fully appreciated only a bit later in
the course. It should, however, make intuitive sense for a finite
number of pairwise disjoint events.
After learning how to operate with events and giving the formal
definition of probability, derived properties of probability are
obtained from the defining axioms.