Transcript PowerPoint

Lecture 8—Probability and Statistics (Ch. 3)
Friday January 25th
•Quiz on Chapter 2
•Classical and statistical probability
•The axioms of probability theory
•Independent events
•Counting events
Reading:
All of chapter 3 (pages 52 - 64)
Homework 2 due TODAY
***Homework 3 due Fri. Feb. 1st****
Assigned problems, Ch. 3: 8, 10, 16, 18, 20
Homework assignments available on web page
Exam 1: two weeks from today, Fri. Feb. 8th (in class)
Classical
Thermodynamics
Classical and statistical probability
Classical probability:
•Consider all possible outcomes (simple events) of a process
(e.g. a game).
•Assign an equal probability to each outcome.
Let W = number of possible outcomes (ways)
Assign probability pi to the ith outcome
1
pi 
W
&
1
i pi  W  W  1
Classical and statistical probability
Classical probability:
•Consider all possible outcomes (simple events) of a process
(e.g. a game).
•Assign an equal probability to each outcome.
Examples:
Coin toss:
W=2
pi = 1/2
Classical and statistical probability
Classical probability:
•Consider all possible outcomes (simple events) of a process
(e.g. a game).
•Assign an equal probability to each outcome.
Examples:
Rolling a dice:
W=6
pi = 1/6
Classical and statistical probability
Classical probability:
•Consider all possible outcomes (simple events) of a process
(e.g. a game).
•Assign an equal probability to each outcome.
Examples:
Drawing a card:
W = 52
pi = 1/52
Classical and statistical probability
Classical probability:
•Consider all possible outcomes (simple events) of a process
(e.g. a game).
•Assign an equal probability to each outcome.
Examples:
FL lottery jackpot:
W = 20M
pi = 1/20M
Classical and statistical probability
Statistical probability:
•Probability determined by measurement (experiment).
•Measure frequency of occurrence.
•Not all outcomes necessarily have equal probability.
•Make N trials
•Suppose ith outcome occurs ni times
 ni 
pi  lim  
N  N
 
Classical and statistical probability
Statistical probability:
•Probability determined by measurement (experiment).
•Measure frequency of occurrence.
•Not all outcomes necessarily have equal probability.
Example:
 ni 
pi  lim    0.312
N  N
 
Classical and statistical probability
Statistical probability:
•Probability determined by measurement (experiment).
•Measure frequency of occurrence.
•Not all outcomes necessarily have equal probability.
More examples:
Classical and statistical probability
Statistical probability:
•Probability determined by measurement (experiment).
•Measure frequency of occurrence.
•Not all outcomes necessarily have equal probability.
More examples:
Statistical fluctuations
 N
0.0
1/ 2
log    a log  N   b
-0.5
log( )
-1.0
-1.5
-2.0
-2.5
-3.0
a  0.516
N

1
0.5
10
0.15
100
0.04
1000 0.0132
10000 0.00356
100000 0.00145
0
1
2
3
log(N)
4
5
The axioms of probability theory
1. pi ≥ 0, i.e. pi is positive or zero
2. pi ≤ 1, i.e. pi is less than or equal to 1
3. For mutually exclusive events, the probabilities
for compound events, i and j, add
pi  j   pi  p j
• Compound events, (i + j): this means either event i occurs, or event
j occurs, or both.
• Mutually exclusive: events i and j are said to be mutually exclusive
if it is impossible for both outcomes (events) to occur in a single
trial.
The axioms of probability theory
1. pi ≥ 0, i.e. pi is positive or zero
2. pi ≤ 1, i.e. pi is less than or equal to 1
3. For mutually exclusive events, the probabilities
for compound events, i and j, add
• In general, for r mutually exclusive events, the probability that one
of the r events occurs is given by:
p  p1  p2  ........  pr
Independent events
Example:
What is the probability of
rolling two sixes?
Classical probabilities:
p6 
1
6
Two sixes:
p6,6  16  16 
1
36
•Truly independent events always satisfy this property.
•In general, probability of occurrence of r independent
events is:
p  p1  p2  ........  pr