probability theory evolved out of practical considerations, i.e. gambling

it’s the science of uncertainty

probability - likelihood or chance of an occurrence of a particular event

basic counting principles

addition principle - method of determining number of elements in a set without actually counting each element

example

M is the set of males in Geog 201 and F is the set of females

since the sets have no members in common the intersection 

M and F are disjoint sets

this doesn’t work if some elements are shared between the sets

example

there are 22 environmental management majors students in Geog 201 and 16 urban development majors with seven of the students pursuing both areas

if M denotes environmental management majors

if U denotes urban development major

you might think that all we need to do is add M and U to get 38 majors but this would be in error

the double majors have been counted twice

the correct answer is to subtract the double majors

= 22 + 16 - 7 =31

so

for any two sets A and B

if the sets are disjoint

example

a survey of firms shows that 750 firms offer employee health insurance, 640 offer dental insurance, and 280 offer both

How many offer health or dental?

H - set of firms offering health

D - set of firms offering dental insurance

so n(H) = 750

n(D) = 640

= 750 + 640 - 280 = 1,110

Venn diagrams

city has 2 newspapers the Sentinel and the Journal; a survey of 100 people shows: 35 people subscribe to the Sentinel, 60 subscribe to the Journal, and 20 to both

a) how many subscribe to the Sentinel but not the Journal

b) how many subscribe to the Journal but not the Sentinel

c) how many do not subscribe to either

can do this in a diagram

or a table

Multiplication principle

start with an example

suppose we have a spinner that can land on 4 numbers 1,2, 3 or 4. Then flip a coin that turns up either heads (H) or tails (T) what are the possible combined outcomes?

so there are 8 possible outcomes

now suppose you asked from the 26 letters of the alphabet how many ways can 3 letters appear in a row on a license plate if no letter is repeated?

To do a tree diagram would be very tedious but we can use multiplication principle (for counting)

If 2 operations O₁ and O₂ are performed in order, with N₁ possible outcomes for the first operation and N₂ for the second, then there are N₁ @ N₂ possible outcomes

this can be generalized to N₁ @ N₂ ..........N_n

permutation

an arrangement of objects in a definite order, the same objects arranged in a different order constitute a different permutation

e.g. ABCD is different than DCBA

permutations (n,r) = # of permutations of n objects r at a time

note 1!=1 0!=1

e.g. 3 objects ABC taken 2 at a time

possibilities AB, AC, BA, CA, BC, CB = 6

P(3,2) = 3!/(3-2)! = (3 x 2 x 1)/1 = 6/1 = 6

combinations- an arrangement of objects made without regard to the order of objects

e.g. ABCD is the same as DCBA

# of combination # # of permutations

e.g. c(3,2) = 3/2!(3-2)! = (3x2x1)/(2x1)x1=6/2=3

ab/ba

ac/ca

bc/cb

probability - likelihood or chance of an occurrence of a particular event

an event must have at least 2 outcomes

examples of events event context possible outcomes

coin toss experiment heads/tails

flooding time flood/no flood

retail store space present/absent

2 definitions of probability

1) objective / a priori - probabilities generated from objective reasoning

e.g. throwing a die and getting a 2 = # of successful outcomes/# of possible outcomes = 1/6 =.1666

2) subjective / a posterior / relative - probabilities generated from experimentation/ history/ empirical study

# of successes # of occurrences 23 times get a 2

# of trials # of events 120 die throws = .1916

this is a practical way of testing objective reasoning about probability of events

example - a die was tossed by Wolf 20,000 times with the following frequencies - was the die fair?

in geography it is very difficult to generate objective probabilities of events, most probabilities are subjectively derived

e.g. # of earthquakes 2

# of years of data 100 = 0.02

if we want expect probabilities to be accurate then we need a very long run of historical data, in other words we want to generate probability statements based on a large set of observations

as the number of trials increases, the proportion of occurrences approaches a limiting value or probability

predictability: although we can establish probabilities for events that doesn't mean we know whether or not a particular outcome will occur

we may know that there is only a 2% chance of an earthquake in any year but we don't know what year it will occur

this is because earthquakes are produced by a stochastic process

but probability is very useful in planning for catastrophic events

P =0 - event never occurs P=1 event always occurs

addition law - used to compute probability of an event A or B

if A and B are outcomes of a given event and are- then the probability of EITHER outcome is the sum of their individual probabilities

e. coin toss A=heads B=tails

addition law P(A or B) = P(A) + P(B) = 0.5+0.5 = 1

e.g. flood A=no flood

B= flood of 5cm

C=flood of 10cm

P(A or B or C) = P(A) + P(B) + P(C) = 0.7+0.2+0.1 = 1

General theorem P(A) + P(B) -P(A and B)

e.g. for 1000 households, where do you shop, what mode?

suburbs city centre

mode=car 150 650 800

mode=walk 150 50 200

total 300 700 1000

probability of suburban trip 300/1000 = 0.30

probability of trip by car = 800/1000 = 0.80

probability of A or B P(A or B) = 0.30+0.80 = 1.1

ERROR!!!

probability of suburban trip and probability of trip by car (P(A), P(B)) are NOT independent events/mutually exclusive events

subtract the probability of a suburban trip and probability of a car trop

e.g. P(A or B) =(0.30+.80-0.15)=.95

if A and B are 2 possible outcomes of events that are independent then:

P(A and B) = P(A) x P(B) (multiplication rule)

example

probability of a crop loss in a given year =0.16

what is the probability of crop losses in 2 successive years

the events (crop loss in a year) are independent of each other

e.g.. dice throw of a 12 P(first 6 and 2nd 6) = 1/6 x1/6 =1/36 = .02777

probability of an event occurring = P(a) then the probability of A not occurring is 1 - P(A)

what if events A and B are not mutually exclusive

P(A and B) = P(A/B) x P(B)

e.g. order events

P(wheat in area 1) = 1/3 =0.333

P(corn in area 2, given wheat in are a 1) =1/2 = 0.5

P( corn and wheat) = P(corn given wheat) x P(wheat)

A = corn, B=wheat P(A and B) = 0.5 x 0.333 = 0.1665

example

suppose there are 50 cereal boxes on a shelf

30 are red and 20 are blue

pick a random sample of 2 boxes without replacement

what is the chance the second box in the sample is blue?

2 ways to pick a sample so that the blue box is the second item

a) 2 blue boxes

b) 1 red and 1 blue

probability of 2 blue is

P(1st pick blue) x P(pick blue if 1st is blue)

1st pick blue is simply 20 out of 50

pick blue if 1st is blue is the conditional probability based on picking blue first

P = (2/5)(19/49)

probability of red then blue

P(1st pick red) is 30 out of 50

P(pick blue if 1st pick red) = 20/49

P(1st pick blue) x P(2nd pick blue if 1st blue) + P(1st pick red) x P(2nd pick blue if 1st red) = (2/5)(19/49) + (3/5)(20/49) = 2/5 or .40

another example using boxes covered with construction paper

problem 1: given 9 boxes, 5 orange 4 green, what is the probability the second box you pick will be green?

2 outcomes

a) 1st pick red then green 5/9 x 1/2 =5/18

b) 1st pick green then pick green 4/9 x 3/8 = 12/72

total probability is a) + b) = 5/18 + 1/6 = 8/18 = 4/9

problem 2: add 1 yellow box to sample to make 10 boxes

now what is probability of getting 2nd box green?

2 possible ways to solve

1 way

3 outcomes

a) 1st orange then green 1/2 x 4/9 = 4/18

b) 1st yellow then green 1/10 x 4/9 = 4/90

c) 1st green then green 2/5 x 1/3 = 2/15

common denominator is 180

so 40/180 + 8/180 + 24/180 = 72/180 = 2/5

2nd way

treat boxes a green or not green

a) not green 1st then green 3/5 x 4/9 = 12/45

b) green 1st then green 2/5 x 1/3 = 2/15

common denominator is 45

so 12/45 + 6/45 = 18/45 = 2/5