Lecture 1: Probability and Counting

History of probability

• Mosteller-Wallace, The Federalist Papers

• Goverment : IQSS

• Finace : STAT 123

• Gambling : Fermat-Pascal 1650's

• Life : Statistics is the logic of uncertainty

Sample Space 표본 공간

• A sample space is the set of all possible outcomes of an experiment.

• An event is a subset of the sample space.

• 통계학은 때로, 직관에 반하는 일 중 하나이므로 엄밀한 수학적 정의가 필요하다.

• 어떠한 사건을 표본 공간에 대한 특정 집합으로 보는 연습이 되어야 한다.

• Naive definition of probability

  $P(A) = \frac{\#\;of\;favorable\;outcomes}{\#\;of\;possible\;outcomes}$

  $$

• Assumes : all outcomes equally likely finite sample space.

  • ex. Dice

Counting

• Multiplication Rule 곱셈 규칙
  : If you have experiment with $n_1$ possible outcomes, and for each outcome of 1st experiment ther are outcomes for 2nd expt ..., for each there are $n_r$ outcomes for $r$th expt, then $n_1$, $n_2$, $n_r$ overall possible outcomes.
  • ex1. Ice cream : 2 corns and 3 flavors = 6 choices
  • ex2. Prob. of full house in poker, 5 cards hand

• Binomial coefficient 이항 계수

  $\begin{pmatrix}n\\k\\ \end{pmatrix} = \frac{n!}{(n-k)!\;k!}, \;0\;if\;k\;>\;n$

  $$

• # subsets of size k, of group of n people

  $\frac{n * (n-1) * (n-2) ... (n-k+1)}{k!} = \frac{n!}{(n-k)!\;k!}$

Sampling table : choose k objects out of n.

	order matter	order doesn't matter
replace	$n^k$	$\begin{pmatrix}n+k-1\\k\\ \end{pmatrix}$
don't replace	$n * (n-1) ... (n-k+1)$	$\begin{pmatrix}n\\k\\ \end{pmatrix}$

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• 출처 : Statistics 110, boostcourse