Lecture 8—Probability and Statistics Lecture 8—Probability and Statistics (Ch. 3)(Ch. 3)

Friday January 25Friday January 25thth

•Quiz on Chapter 2

•Classical and statistical probability

•The axioms of probability theory

•Independent events

•Counting events

Reading: Reading: All of chapter 3 (pages 52 - 64)All of chapter 3 (pages 52 - 64)Homework 2 due TODAYHomework 2 due TODAY***Homework 3 due Fri. Feb. 1st*******Homework 3 due Fri. Feb. 1st****Assigned problems, Assigned problems, Ch. 3Ch. 3: 8, 10, 16, 18, : 8, 10, 16, 18,

2020Homework assignments available on Homework assignments available on

web pageweb pageExam 1: two weeks from today, Fri. Feb. 8th (in Exam 1: two weeks from today, Fri. Feb. 8th (in class)class)

ClassicalClassicalThermodynamicsThermodynamics

Classical and statistical probabilityClassical 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 1& 1i i

i

p p WW W

Classical and statistical probabilityClassical 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:Coin toss:

WW = 2 = 2 ppii = 1/2 = 1/2

Classical and statistical probabilityClassical 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:Rolling a dice:

WW = 6 = 6 ppii = 1/6 = 1/6

Classical and statistical probabilityClassical 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:Drawing a card:

WW = 52 = 52 ppii = 1/52 = 1/52

Classical and statistical probabilityClassical 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:FL lottery jackpot:

WW = 20M = 20M ppii = 1/20M = 1/20M

Classical and statistical probabilityClassical and statistical probability

Statistical probability:

•Probability determined by measurement (experiment).

•Measure frequency of occurrence.

•Not all outcomes necessarily have equal probability.•Make Make N N trialstrials

•Suppose Suppose iithth outcome occurs outcome occurs nnii times times

lim ii N

np

N

Classical and statistical probabilityClassical and statistical probability

Statistical probability:

•Probability determined by measurement (experiment).

•Measure frequency of occurrence.

•Not all outcomes necessarily have equal probability.Example: lim 0.312i

iN

np

N

Classical and statistical probabilityClassical 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 probabilityClassical and statistical probability

Statistical probability:

•Probability determined by measurement (experiment).

•Measure frequency of occurrence.

•Not all outcomes necessarily have equal probability.More examples:

0 1 2 3 4 5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

N 1 0.510 0.15100 0.041000 0.013210000 0.00356100000 0.00145

log( )

log(N)

log log

0.516

a N b

a

Statistical fluctuationsStatistical fluctuations

1/ 2N

The axioms of probability theoryThe 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

i ji jp p p

• Compound events, (Compound events, (ii + + jj): this means either event ): this means either event ii occurs, or event occurs, or event jj occurs, or both. occurs, or both.

• Mutually exclusive: events Mutually exclusive: events ii and and jj are said to be mutually exclusive are said to be mutually exclusive if it is impossible for both outcomes (events) to occur in a single if it is impossible for both outcomes (events) to occur in a single trial.trial.

The axioms of probability theoryThe 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 In general, for rr mutually exclusive events, the probability that one mutually exclusive events, the probability that one of the of the rr events occurs is given by: events occurs is given by:

1 2 ........ rp p p p

Independent eventsIndependent events

Example:What is the probability of What is the probability of rolling two sixes?rolling two sixes?

Classical probabilities:Classical probabilities:

16 6p

Two sixes:Two sixes:

1 1 16,6 6 6 36p

•Truly independent events always satisfy this property.

•In general, probability of occurrence of r independent events is:1 2 ........ rp p p p