8/25/2017

## Example 2.2 Revisit

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?

## Definitions

• Mutually Exclusive(disjoint): $A\cap B = \emptyset$
• Exhaustive: $A \cup B = S$
• Partition:
• $$A\cap B = \emptyset$$
• $$A \cup B = S$$

## Definition of Conditional Probability

• The conditional probability of event $$A$$, given event $$B$$ is written as $$P(A|B)$$

• In general, if event $$B$$ has nonzero probability ($$P(B) > 0$$), then the conditional probability is defined as $P(A|B) = \frac{P(A\cap B)}{P(B)}$

## Example 3.1

Roll a pair of dice. Given that the two dice have different values, find the probability that the sum of the dice is an even number.

## Example 3.2

We will toss a fair coin 3 times. Let event:

• A = exactly 2 heads;
• B = $$1_{st}$$ toss is heads;
• C = $$2_{nd}$$ toss is heads

Find $$P(A), P(B), P(A\cap B), P(B|A), P(A|B)$$ and $$P(B|C)$$

## Another Useful form

$P(A|B) = \frac{P(A\cap B)}{P(B)} = \frac{\frac{N(A\cap B)}{N(S)}}{\frac{N(B)}{N(S)}} = \frac{N(A\cap B)}{N(B)}$

## Distributive Laws

For events $$A_1$$, $$A_2$$ and $$B$$

1. $$(A_1 \cup A_2) \cap B = (A_1 \cap B) \cup (A_2 \cap B)$$
• Distributive laws of multiplication $(A_1 + A_2) \times B = A_1 \times B + A_2 \times B$
2. $$(A_1 \cap A_2) \cup B = (A_1 \cup B) \cap (A_2 \cup B)$$

## Distributive Laws

For any events $$A_1, A_2, \cdots$$

1. $$(\bigcup_i A_i) \cap B = \bigcup_i(A_i \cap B)$$

2 .$$(\bigcap_i A_i) \cup B = \bigcap_i(A_i \cup B)$$

## Recap

• Definitions: Random Experiment, Potential Outcome, Element $$\omega$$, Event, empty set $$\emptyset$$, sample space $$S$$, Union $$\cup$$, Intersection $$\cap$$

• Probability Rules
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 $$P(\bigcup_{i=1}^n A_i) = \sum_{i=1}^\infty P(A_i)$$
• DeMorgan's law
1. $$(A\cup B)^c = A^c \cap B^c$$
2. $$(A\cap B)^c = A^c \cup B^c$$
• Conditional Probability: $$P(A | B) = \frac{P(A\cap B)}{P(B)}$$