Download Chapters 13 and 14 Notes

Introduction to
Probability
(Dr. Monticino)
Assignment Sheet
 Read Chapters 13 and 14
 Assignment #8 (Due Wednesday March 23rd )
 Chapter 13
 Exercise Set A: 1-5; Exercise Set B: 1-3
 Exercise Set C: 1-4,7; Exercise Set D: 1,3,4
 Review Exercises: 2,3, 4,5,7,8,9,11

Chapter 14
 Exercise Set A: 1-4; Exercise Set B: 1-4, 5
 Exercise Set C: 1,3,4,5; Exercise Set D: 1 (just calculate
probabilities)
 Review Exercises: 3,4,7,8,9,12
Overview
 Framework
 Equally likely outcomes
 Some rules
Probability Framework
 The sample space, , is the set of all outcomes
from an experiment
 A probability measure assigns a number to each
subset (event) of the sample space, such that



0  P(A)  1
P( ) = 1
If A and B are mutually exclusive (disjoint) subsets,
then P(A  B) = P(A) + P(B) (addition rule)
Equally Likely Outcomes
 Outcomes from an experiment are said
to be equally likely if they all have the
same probability.
 If
there are n outcomes in the experiment
then the outcomes being equally likely
means that each outcome has probability
1/n
 If there are k outcomes in an event, then
the event has probability k/n
 “Fair” is often used synonymously for
equally likely
Examples
 Roll a fair die
 Probability of a 5 coming up
 Probability of an even number coming up
 Probability of an even number or a 5
 Roll two fair die
 Probability both come up “1” (double ace)
 Probability of a sum of 7
 Probability of a sum of 7 or 11
More Examples
 Spin a roulette wheel once
 Probability of “11”
 Probability of “red”; probability of “black”;
probability of not winning if bet on “red”
 Draw one card from a well-shuffled deck of
cards



Probability of drawing a king
Probability of drawing heart
Probability of drawing king of hearts
Conditional Probability
 All probabilities are conditional
 They are conditioned based on the
information available about the experiment
 Conditional probability provides a
formal way for conditioning
probabilities based on new information
 P(A
| B) = P(A  B)/P(B)
 P(A  B) = P(A | B)  P(B)
Multiplication Rule
 The probability of the intersection of
two events equals the probability of the
first multiplied by the probability of the
second given that the first event has
happened
 P(A
 B) = P(A | B)  P(B)
Examples
 Suppose an urn contains 5 red marbles
and 8 green marbles
 Probability
of red on first draw
 Red on second, given red on first (no
replacement)
 Red on first and second
Independence
 Intuitively, two events are independent if
information that one occurred does not affect
the probability that the other occurred
 More formally, A and B are independent if



P(A | B) = P(A)
P(B | A) = P(B)
P(AB) = P(A)P(B)
(Dr. Monticino)