i - probabilityfrst231-forestry.sites.olt.ubc.ca/files/2012/07/probability-part1.pdf · section 2 -...

Section 2 - Probability (1)FRST 231 / WOOD 242Introduction to Biometrics & Business Statistics

I - Probability

What is Probability?– the chance of an event occuring

eg

�classical probability

�empirical probability

�subjective probability


I - Probability

Probability - Terminology

– random (probability) experiments• a process that leads to well defined

results called outcomes…

– outcomes• the result of a single trial of an

experiment

• aka: elements, sample points

eg

experiment:

outcome:


I - Probability

Probability - Terminology– sample space

• the set of all possible outcomes in a random experiment

• a listing of all sample points

eg - flip a coin twice

– event• a subset of the sample space with

some special characteristics

eg - get a head first

- get at least one tail


I - Probability

Probability - Terminology– Venn diagram

• a graphical representation of probability problems

A B

S

A = (HH, HT)

B = (HT, TH, TT)


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1. classical probability– uses sample spaces to determine

the numerical probability that an event will happen

– a sample space has ‘n’ equally likely outcomes; an event contains ‘f’ of those

ie

eg

( )nf

EP =


I - Probability

2. empirical probability– uses observations (frequency

distributions) to determine numerical probabilities

eg

• flip a coin 500 times...

P(head) =

• frequency distributions...

P(belonging to a class) =


I - Probability

3. subjective probability– educated guess, opinions, inexact

information

– based on a person’s knowledge, experience, or evaluation of a situation

eg


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3. subjective probability

– odds

• if E is an event...

- the odds in favour of E are:

- the odds against E are:

( ))(

)(

EPEP

Eodds =

( ))()(

EPEP

Eodds =


I - Probability

Laws of Probability

�all outcomes in a sample space are equally likely to occur (ie, each has the same probability)

where:

( ) α=EP

10 ≤≤= αα


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Laws of Probability�an event’s complement is when

that event does not occur

E_E

S

( ) ( ) ( ) 1==+ SPEPEP

( ) ( )EPEP =−∴ 1

eg B = (HT, TH, TT)


I - Probability

Probability - Compound Events

– experiment: roll a dieS = (1, 2, 3, 4, 5, 6)

– 3 events: roll an even number

A = (2, 4, 6)

roll less than 3

B = (1, 2)

roll a 1

C = (1)

S

B A

C


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S

B A

intersection of events:

•probability of events occurring simultaneously, (overlapping elements)

( ) ( )BPAPBAP ×=∩ )(eg

“multiplication rule for independent events”


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S

A

C

mutually exclusive or disjoint events:

•occurrence of one event precludes the other, no intersection (no common elements)


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S

A

C

union of events:

•probability of one event or another (others) occurring

eg

“addition rule for mutually exclusive events”

( ) ( ) ( )CPAPCAP +=∪


I - Probability


S

B A

union of events that are not mutually exclusive:

eg A = (2, 4, 6) P(A) = 1/2

B = (1, 2) P(B) = 1/3


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S

B A

union of events that are not mutually exclusive:

eg

“addition rule for events that are not mutually exclusive”

( ) ( ) ( ) ( )BAPBPAPBAP ∩−+=∪

( ) ( ) ( ) ( )BPAPBPAP −+=


I - Probability


Review:

P (Queen of Hearts) =

P (Queen of Hearts or Queen of Spades) =

P (a Queen or a Heart) =


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Laws of Probability (review)

�general formula

�complement law

( )nf

EP =

10 ≤≤= αα

= α

where

( ) ( ) ( ) 1==+ SPEPEP

( ) ( )EPEP =−∴ 1


I - Probability

Laws of Probability (review)

�addition law(for mutually exclusive events)

�generalized addition law(events need not be mutually exclusive)

( ) ( ) ( )BPAPBAP +=∪

( ) ( ) ( ) ( )BAPBPAPBAP ∩−+=∪

( ) ( ) ( ) ( )BPAPBPAP −+=


I - Probability

Laws of Probability

�multiplication law(for independent events)

( ) ( )BPAPBAP ×=∩ )(

independent events - the probability of one event occurring does not effect the probability of the other occurring.

dependent events - the outcome of the second event is affected in such a way that the probability is changed.


I - Probability

Laws of Probability

�generalized multiplication law(events need not be independent)

( ) ( )ABPAPBAP ×=∩ )(

conditional probability

the probability that event B occurs after (given that) event A has already occurred.

if events A and B are independent, then...

P(B|A) = P(B)


I - Probability

Probability

Review: - drawing from a deck of cardswithout replacement

P (Q, K) =

P (Q, Q) =

P (Q, Q, Q) =

P (J, Q, K) =

P (Heart, Diamond) =

P (Heart, Heart, Heart) =


I - Probability

Probability

log sort problem:

A mill purchases logs from 2 suppliers:

companies loggers80% of logs 20% of logs1% defective 2% defective

If all of the logs are randomly put in a log sort, what is the probability that a randomly selected log will be defective?


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Probability

log sort problem (cont.):


I - Probability

Probability



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Conditional Probability

eg flip a coin twice S = (HH, HT, TH, TT)

2 events: at least one Head: A = (HH, HT, TH)first toss a Head: B = (HH, HT)

What is P(B|A)?

( ) ( )ABPAPBAP ×=∩ )(

( ) ( )( )AP

BAPABP

∩=


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log sort problem:

1,000 logs:

company logger TOTALdefective 8 4 12not defective 792 196 988TOTAL 800 200 1000

What is the probability of selecting a defective log given that it came from a logger?


I - Probability




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Conditional Probability– has order, depending on the

question being asked...

�natural order- ordered in terms of events

�Bayesian probability- reverse order- based on some information, what is the

probability of an event which occurred in the past?

eg P (D|C)

ie P (B|A) - B occurs after A

eg P (C|D)


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�Bayesian probability (cont.)- solve using conditional probability


I - Probability


species blow-down problem:

species: blow-down rate:

D-fir 0.0001cedar 0.05hemlock 0.01

Given a tree that has blown down, what is the probability that it is a cedar?


I - Probability


species blow-down problem (cont.):


I - Probability


�Bayesian probability (cont.)

Bayes’ Theorem: for 2 events, A and B, where B follows A:

( ) ( )( )∑

=

∩

∩=

n

jj

jj

BAP

BAPBAP

1

( ) ( )( ) ( )∑

=

=n

jjj

jj

ABPAP

ABPAP

1

i - probabilityfrst231-forestry.sites.olt.ubc.ca/files/2012/07/probability-part1.pdf · section 2 -...

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