8/23/2017

## In Lecture 1

• Potential Outcome: One specific result from a random experiment

• Element $$\omega$$: a single potential outcome

• Event: a collection of some outcomes
• empty set $$\emptyset$$: no outcomes
• sample space $$S$$ or $$\Omega$$: consists of all outcomes
• Union $$\cup$$: a new set that contains each outcome found in an of the component events
• Intersection $$\cap$$: a new set that contains only the outcomes found in all of the component event.

## Disjoint

• A pair of events A, B is disjoint, if they have no outcome in common

• $$A\cap B = \emptyset$$

• Also called mutually exclusive

• A collection of event is disjoint if every pair of events is disjoint

## Three Probability Axioms

1. For each event A, $0\le P(A) \le 1$
2. For the sample space S, $P(S) = 1$
3. If $$A_1, A_2, \cdots$$ is a collection of disjoint events, then $P(\bigcup_{i=1}^n A_i) = \sum_{i=1}^\infty P(A_i)$

## Two Thoerem Based on Probability Axioms

1. The probability of the empty set $$\emptyset$$ is always 0

2. If $$A_1, A_2, \cdots, A_n$$ is a collection of finitely many disjoint events, then the probability of the union of the events equals the sum of the probabilities of the events: $P(\bigcup_{i=1}^n A_i) = \sum_{i=1}^n P(A_i)$

## Equally Likely Events

• Revisit on rolling a die

• Theorem: If s sample space S has n equally likely outcomes, then each outcome has probability $$\frac{1}{n}$$ of occuring

• Corollary: If sample space S has n equally likely outcomes, and A is an event with j outcomes, then event A has probability $$\frac{j}{n}$$ of occuring, i.e., $P(A) = \frac{j}{n}$

• Corollary If sample space S has a finite number of equally likely outcomes, then event A has probability $P(A) = \frac{|A|}{|S|}$

## DeMorgan's law

1. $(A\cup B)^c = A^c \cap B^c$

2. $(A\cap B)^c = A^c \cup B^c$

• $P(A\cup B) = P(A) + P(B) - P(A\cap B)$

## Example 1

Three of the major commercial computer operating systems are Windows, Mac OS, and Red Hat Linux Enterprise. A Computer Science professor selects 50 of her students and asks which of these three operating systems they use. The results for the 50 students are summarized below.

• 30 students use Windows
• 16 students use at least two of the operating systems
• 9 students use all three operating systems
• 18 students use Mac OS
• 46 students use at least one of the operating systems
• 11 students use both Windows and Linux
• 11 students use both Windows and Mac OS

## Example 1 continued

Let Windows = W, Mac OS = M, and Red Hat Linux Enterprise = L

1. $$N(W^c \cap M^c)$$
2. $$P(W^c \cup M^c)$$
3. $$N(W \cup M \cup L)$$

## Example 2

In a certain population, 10% of the population are rich, 5% are famous, and 3% are both.

• What is the probability a randomly chosen person is not rich?
• What is the probability a randomly chosen person is rich but not famous?
• What is the probability a randomly chosen person is either rich or famous?
• What is the probability a randomly chosen person is either rich or famous but not both?
• What is the probability a randomly chosen person has neither wealth nor fame?