Upload
marc-nebb
View
218
Download
0
Embed Size (px)
Citation preview
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