lecture4 (1).pdf

Lecture 4

Zhihua (Sophia) Su

University of Florida

Jan 21, 2015

STA 4321/5325 Introduction to Probability 1

Agenda

Conditional ProbabilityIndependence

Reading assignment: Chapter 2: 2.7

STA 4321/5325 Introduction to Probability 2

Conditional Probability

Many times, some partial information about the outcome of arandom experiment is available and we want to revise ourestimates of the chance of an event accordingly.

STA 4321/5325 Introduction to Probability 3

Conditional Probability

DefinitionLet A and B be two events in a random experiment with samplespace S. Then, the probability of the event A given that theevent B has occurred is defined as

P (A | B) ≡ P (A ∩B)

P (B)

provided that P (B) > 0.

The symbol P (A | B) is read “THE PROBABILITY of AGIVEN B”.

STA 4321/5325 Introduction to Probability 4

Conditional Probability

Example: Toss a fair die. Let A denote the event that theoutcome is 2, 4 or 6. If somebody asks you to play the followinggame:

“If event A occurs, you pay me $10, otherwise I will pay you$10,”

will you play the game?

STA 4321/5325 Introduction to Probability 5

Conditional Probability

Example (cont’d): Suppose the die is cast in a secret chamberand you have a helpful source who tells you that the result was4, 5 or 6. You have a chance to withdraw or remain in thegame. What would you do?

STA 4321/5325 Introduction to Probability 6

Conditional Probability

Conditional probabilities satisfy the 3 axioms of probability.

ResultLet B be an event with P (B) > 0. Then

1. 0 ≤ P (A | B) ≤ 1 for every event A.

2. P (S | B) = 1.

3. If A1, A2, A3, . . . are mutually exclusive events, then

P

( ∞⋃i=1

Ai | B

)=

∞∑i=1

P (Ai | B).

STA 4321/5325 Introduction to Probability 7

Independence

If the extra information provided by knowing that an event Bhas occurred does not change the probability of A, i.e., ifP (A | B) = P (A), then the events A and B are said to beindependent.

DefinitionTwo events A and B are said to be independent if

P (A ∩B) = P (A)P (B).

STA 4321/5325 Introduction to Probability 8

Independence

P (A ∩B) = P (A)P (B) is equivalent to stating thatP (A | B) = P (A) or P (B | A) = P (B), if the conditionalprobability P (A | B) or P (B | A) exist.

Multiplicative rule: If A1, A2, . . ., An are n events, then

P (A1 ∩A2 ∩ · · · ∩An) = P (A1)× P (A2 | A1)

×P (A3 | A1 ∩A2)

· · · × P (An−1 | A1 ∩A2 ∩ · · · ∩An−2)

×P (An | A1 ∩A2 ∩ · · · ∩An−1).

STA 4321/5325 Introduction to Probability 9

Independence

Example: Draw a card from a shuffled 52 card deck with nopreference to any card. Let A denote the event that a king isdrawn, and let B denote the event that a diamond is drawn.Are events A and B independent?

STA 4321/5325 Introduction to Probability 10