Havard Stat 110 Lecture 2

Story Proofs, Axioms of Probability

Some hints

Some hints and comments:

00:09 Don't lose your common sense (And not only rely on them).
02:38 Do check answer (By doing simple and extreme cases).
04:05 Label people and objects.

Order does not matter with replacement

13:15

Pick $k$ times from set of $n$ objects where order does not matter with replacement.

There are about $\binom{n+k-1}{k}$ ways to pick.

22:40 Imagine put $k$ balls into $n$ boxes.

29:30 Extra: Bose-Einstein condensate (Bose-Einstein 凝聚态)

Story Proof

31:16

Example 1

Use story proof to prove $\displaystyle\binom{n}{k} = \binom{n}{n-k}$

This is easy as we can consider divide $n$ people into two teams: one contains $k$ people and the other one contains $n-k$ people.

32:14

example 2

Prove
$$ n\binom{n-1}{k-1} = k\binom{n}{k} $$

Consider the story below:

Pick $k$ people out of $n$ , with $1$ designated as President.

The solution is obviously

\[ k\binom{n}{k} \]

And transform the view of the problem:

First choose $1$ President, which has $n$ ways.
Then choose $k-1$ people from $n-1$ people.

35:06

Example 3 (Vander Monde)

Prove
$$ \binom{m+n}{k} = \sum\limits_{j=0}^k \binom{m}{j}\binom{n}{k-j} $$

The left annotation means: choose $k$ people from $m+n$ people.

Then we consider divide $m+n$ people into two groups: first contains $m$ people and second contains $n$ people. Next choose $j$ people from first one and $k-j$ people second one.

Non-naive Probability Axioms

39:15 Non-naive definition

Non-naive Definition of Probability

A Probability Sample consists of $S$ and $P$, where $S$ is a sample space and $P$ is a function which takes an event $A\subset S$ as input, returns $P(A)\in [0,1]$ as outputs such that:

$P(\varnothing) = 0$ , $P(S)=1$ .
$\displaystyle P\left(\bigcup_{n=1}^\infty A_n\right) = \sum\limits_{n=1}^\infty P(A_n)$ if $A_j$ are disjoint.