Mean = 41.21 Median = 42.5 s = 7.59

x - 1s–

x + 1s–

Assignment #1

Course Schedule

Probabilities in Geography

• The analyses of many problems (daily or

geographic) are often based on probabilities, such

as:

• What are the “chances” of having rain over the

weekend?

• What is the “likelihood” that the 100-year flood will

occur within the next ten years?

• How “likely” is it that a pixel on a satellite image is

correctly classified or misclassified?

Probability & Probability Distribution

• We summarize a sample statistically and want to make some inferences about the population (e.g., what proportion of the population has values within a given range)

• The concept of probability is the key to making statistical inferences by sampling a population

• What we are doing is trying to ascertain the probability of an event having a given outcome

• This requires us to be able to specify the distribution of a variable before we can make inferences

Probability & Probability Distributions

• Previously, we looked at some proportions of area under the normal curve:

Source: Earickson, RJ, and Harlin, JM. 1994. Geographic Measurement and Quantitative Analysis. USA: Macmillan College Publishing Co., p. 100.

Probability & Probability Distributions

• BUT before we could use the normal curve, we have to find out if this is the right distribution for our variable …

• While many natural phenomena are normally distributed, there are other phenomena that are best described using other distributions

• Background on probabilities (terminology & rules), and a few useful distributions:

• Discrete distributions: Binomial and Poisson

• Continuous distributions: Normal and its relatives

Probability-Related Concepts

• An event – Any phenomenon you can observe that can have more than one outcome (e.g., flipping a coin)

• An outcome – Any unique condition that can be the result of an event (e.g., flipping a coin: heads or tails), a.k.a simple event or sample points

• Sample space – The set of all possible outcomes associated with an event

– e.g., flip a coin – heads (H) and tails (T)

– e.g., flip a coin twice – HH, HT, TH, TT

• Associated with each possible outcome in a

sample space is a probability

• Probability is a measure of the likelihood of

each possible outcome

• Probability measures the degree of uncertainty

• Each of the probabilities is greater than or equal

to zero, and less than or equal to one

• The sum of probabilities over the sample space

is equal to one

Probability-Related Concepts

Probability – Examples

• Example I – Flip a coin

– Two possible outcomes: “heads”, “tails”

– Each outcome is equally likely

– “heads” and “tails” have the same probability

(0.5)

– The sum of probabilities over the sample

space is one

– # of “heads” and # of “tails” will be nearly equal

Probability – Examples

• Example II – Flip a coin twice– Four outcomes are equally likely

– Tosses of the coin are independent

– Each outcome has probability 1/4

– The probability of a head on Flip 1 and a head on Flip 2 is 1/2 * 1/2 = 1/4

Outcome First flip Second flip

1 Heads Heads

2 Heads Tails

3 Tails Heads

4 Tails Tails

How To Assign Probabilities to Experimental Outcomes?

• There are numerous ways to assign probabilities to the elements of sample spaces

• Classical method assigns probabilities based on the assumption of equally likely outcomes

• Relative frequency method assigns probabilities based on experimentation or historical data

• Subjective method assigns probabilities based on the assignor’s judgment or belief

Classical Method

• This approach assumes that each outcome is

equally likely

• If an experiment has n possible outcomes, this

method would assign a probability of 1/n to each

outcome.

• It is an appropriate way to assign probabilities to

the outcomes in special kinds of experiments

Classical Method

• Example I: Rolling a die

• Sample Space: S = {1, 2, 3, 4, 5, 6}

• Probabilities: Each sample point has a 1/6

chance of occurring.

Classical Method

• Example II – Flip four coins– Let “0” represent “heads” and “1” represents “tails”– For each toss, the probability of “heads” or “tails” is ½– Assuming that outcomes of the four tosses are

independent from one another– Sixteen possible outcomes

× × × ×

½ ½ ½ ½ Probability of each outcome:

½ * ½ * ½ * ½ = 1/16 = 0.0625

0000 0100 1000 1100

0001 0101 1001 1101

0010 0110 1010 1110

0011 0111 1011 1111

Relative Frequency Method

• The second way is to assign them on the basis of relative frequencies

• Example

– Given a weather pattern, a meteorologist may note that in 65 out of the last 100 times that such a pattern prevailed there was measurable precipitation the next day

– If there were such a weather pattern today, what would the probability of having rain tomorrow be?

– The possible outcomes – rain or no rain tomorrow – are assigned probabilities of 0.65 and 0.35, respectively

Subjective Method

• When extreme weather conditions occur it might be inappropriate to assign probabilities based solely on historical data

• We can use any data available as well as our experience and intuition, but ultimately a probability value should express our degree of belief that the experimental outcome will occur

• The best probability estimates often are obtained by combining the estimates from the classical or relative frequency approach with the subjective estimates.

Probability Rules

• Rules for combining multiple probabilities• A useful aid is the Venn diagram - depicts multiple

probabilities and their relations using a graphical depiction of sets

• The rectangle that forms the area of the Venn Diagram represents the sample (or probability) space, which we have defined above

• Figures that appear within the sample space are sets that represent events in the probability context, & their area is proportional to their probability (full sample space = 1)

A B

Probability Rules• We can use a Venn diagram to describe the

relationships between two sets or events, and the corresponding probabilities

•The union of sets A and B (written symbolically is A B) is represented by the areas enclosed by set A and B together, and can be expressed by OR (i.e. the union of the two sets includes any location in A or B)

•The intersection of sets A and B (written symbolically as A B) is the area that is overlapped by both the A and B sets, and can be expressed by AND (i.e. the intersection of the two sets includes locations in A AND B)

A B

A B

Addition Rule• If sets A and B do not overlap in the Venn diagram,

the sets are disjoint, and this represents a case of two independent, mutually exclusive events

•The union of sets A and B here uses the addition rule, where

P(A = P(A) + P(B)

•You can think of this in terms of areas of the events, where the union in this case is simply the sum of the areas

•The intersection of sets A and B here results in the empty set (symbolized by ), because at no point do the circles overlap

A B

A B

P(A = P(A) + P(B)

P(A =

Probability Rules

• The union of sets A and B here uses the addition rule, where

P(A = P(A) + P(B)

P(A = 2/6 + 2/6

P(A = 4/6 = 2/3 = 0.67

•The outcomes represented here are mutually exclusive, thus there is no intersection between sets A and B, thus P(A =

A B

A B

P(A = P(A) + P(B)

P(A =

• For example, suppose set A represents a roll of 1 or 2 on a 6-sided die, so P(A)=2/6, and set B represents a roll of 3 or 4, so P(B)=2/6

Probability Rules – General Addition Rule

• If sets A and B do overlap in the Venn diagram, the sets are independent but not mutually exclusive

•The union of sets A and B here isP(A = P(A) + P(B) - P(A

because we do not wish to count the intersection area twice, thus we need to subtract it from the sum of the areas of A and B when taking the union of a pair of overlapping sets

The intersection of sets A and B here is calculated by taking the product of the two probabilities, a.k.a. the multiplication rule:

A B

A B

P(A = P(A) * P(B)

P(A = P(A) + P(B) - P(A

General Addition Rule• Consider set A to give the chance of precipitation

at P(A)=0.4 and set B to give the chance of below freezing temperatures at P(B)=0.7

•The intersection of sets A and B here is P(A = P(A) * P(B)

P(A = 0.4 * 0.7 = 0.28

This expresses the chance of snow at P(A = 0.28

•The union of sets A and B here is

P(A = P(A) + P(B) - P(A

P(A = 0.4 + 0.7 – 0.28 = 0.82

This expresses the chance of below freezing temperatures or precipitation occurring at P(A = 0.82

A B

P(A = P(A) + P(B) - P(A

A B

P(A = P(A) * P(B)

Complement• Consider set A to give the chance of precipitation

at P(A)=0.4 and set B to give the chance of below freezing temperatures at P(B)=0.7

•The complement of set A is

P(A’ = 1 - P(A)

P(A’ = 1 – 0.4 = 0.6

This expresses the chance of it not raining or snowing at P(A’ = 0.6

•The complement of the union of sets A and B is

P(A’ = 1 – [P(A) + P(B) - P(AP(A’ = 1 – [0.4 + 0.7 – 0.28] = 0.18

This expresses chance of it neither raining nor being below freezing at P(A’ = 0.18

P(A’ = 1 - P(A)

P(A’ = 1 – [P(A) + P(B) - P(A

A A’

A BP

(A

’

Probability Rules• We can also encounter the situation where set A is

fully contained within set B, which is equivalent to saying that set A is a subset of set B:

• For example, set A might represent precipitation events with >= 5 inches, whereas set B denotes any events with >= 1 inch A is contained with B because anytime A occurs, B occurs as well

• In probability terms, this situation

occurs when outcome B is a

necessary precondition for outcome

A to occur, although not vice-versa

(in which case set B would be

contained in set A instead)

A

B

Probability – Example

• Example – # of malls within cities City # of Malls

A 1 B 4 C 4 D 4 E 2 F 3

• We might wonder if we randomly pick one of these six cities, what is the probability (chance) that it will have n malls?

SampleSpace

Each count of the # of malls in a city is an event

Random Variables

• What we have here is a random variable – defined as a function that associates a unique numerical value with every outcome of an experiment

• To put this another way, a random variable is a function defined on the sample space this means that we are interested in all the possible outcomes

• A random variable X is a rule that assigns a numerical value to each outcome in the sample space of an experiment

Random Variables

• The value of the random variable will vary from trial to trial as the experiment is repeated

• We use an uppercase letter to denote a random variable and a lowercase letter to denote a particular value of the variable

• A random variable can be classified as being either discrete or continuous depending on the numerical values it assumes

Discrete & Continuous Variables

• Discrete variable – A variable that can take on

only a finite number of values– # of malls within cities

– # of vegetation types within geographic regions

– # population

• Continuous variable – A variable that can take

on an infinite number of values (all real number

values)– Elevation (e.g., [500.0, 1000.0])

– Temperature (e.g., [10.0, 20.0])

– Precipitation (e.g., [100.0, 500.0]

Probability Distribution & Probability Function

• The question was: If we randomly pick one of the six cities, what is the probability (or chance) that it will have n malls?

• To answer this question, we need to form a probability function (probability distribution) from the sample space that gives all values of a random variable and their probabilities

• Then we can find the probability that a randomly selected city has n malls from the probability function

Probability Function & Probability Distribution

• The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable

• In other words, a probability distribution expresses the relative number of times we expect a random variable to assume each and every possible value

• The probability distribution of a random variable may be represented by a table, a graph, or an equation

Probability Function & Probability Distribution

• The probability distribution is defined by a probability function, denoted by p(X) or f(x), which provides the probability for each value of the random variable

• p(X) or f(x) represents the probability function or the probability distribution for the random variable X

Probability Function – An Example• Here, the values of xi are drawn from the four

outcomes, and their probabilities are the number of events with each outcome divided by the total number of events:

City # of Malls

A 1

B 4

C 4

D 4

E 2

F 3

xi P(xi)

1 1/6 = 0.1672 1/6 = 0.1673 1/6 = 0.1674 3/6 = 0.5

• The probability of an outcome P(xi) = # of times an outcome occurred

Total number of events

Probability Function

• We can plot this probability distribution as a probability function:

• This plot uses thin lines to denote that the probabilities are massed at discrete values of this random variable

xi p(xi)

1 1/6 = 0.1672 1/6 = 0.1673 1/6 = 0.1674 3/6 = 0.5 0

0.25

0.50

p(x i)

1 2 3 4

xi

Probability Mass Functions

• A discrete random variable can be described by a

probability mass function (pmf)

• A probability mass function is usually represented

by a table, graph, or equation

• The probability of any outcome must satisfy:

0 <= p(X=xi) <= 1 i = 1, 2, 3, …, k-1, k

• The sum of all probabilities in the sample space

must total one, i.e.1)(

1

k

iixXp

Probability Mass Function

• Example: # of malls in cities

• This plot uses thin lines to denote that the probabilities are massed at discrete values of this random variable

xi p(X=xi)

1 1/6 = 0.1672 1/6 = 0.1673 1/6 = 0.1674 3/6 = 0.5

0

0.25

0.50

p(x i)

1 2 3 4

xi

Discrete Probability Distribution

• We can calculate the mean and variance of a discrete probability distribution:

• We use µ and σ2 here because the basic idea of a probability distribution is to use a large number of samples to approach the distribution of a population

xi *p(xi)i=1

i=k

(xi – x)2*p(xi)i=1

i=k

Continuous Random Variables

• Continuous random variable can assume all real

number values within an interval (e.g., rainfall, pH)

• The probability distribution of a random

continuous variable is described by probability

density functions (pdf)

• A probability density function (pdf) is usually

represented by a graph or equation

• Again, there are two fundamental requirements for

a probability density function (pdf):

0)( xf

1)( dxxf

x

area=1

f(x)

µ

Probability Density Functions

• Theoretically, a continuous variable’s range can extend from negative infinity to infinity, e.g. the normal distribution:

• The tails of the normal distribution’s curve extend infinitely in each direction, but the value of f(x) approaches zero, getting closer and closer, but never reaching zero

x

area=1f(x)

• The probability of a continuous random variable X within an arbitrary interval is given by:

• Simply calculate the shaded shaded area if we know the density function, we could use calculus

x

f(x)

a b

b

adxxfbXap )()(

Probability Density Functions

• Fortunately, we do not need to solve the integral ourselves to practice statistics … instead, if we can match the f(x) up to some known distribution, we can use a table of probabilities that someone else has developed

• Tables A.2 through A.6 in the epilogue of the Rogerson text (pp. 214-221) give probability values for several distributions, including the normal distribution and some related distributions used by various inferential statistics

Probability Density Functions

• Suppose we are interested in computing the probability of a continuous random variable at a certain value of x (e.g. at d):

• As the interval from c to d becomes vanishingly narrow, the area below the curve within it becomes vanishingly small

x

f(x)a b

d• Can we find the probability of a value occurring at d? p(d) = ?

p(x)c

d

0 as c d

c

• No, p(d) = 0 … why? The reasons is: