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Correlations Revisited
Probability
I think you're begging the question, said Haydock,
and I can see looming ahead one of those
terrible exercises in probability where six men
have white hats and six men have black hats
and you have to work it out by mathematics how
likely it is that the hats will get mixed up and in
what proportion. If you start thinking about things
like that, you would go round the bend. Let me
assure you of that!
Agatha Christie
The Mirror Crack's
Misunderstanding of probability may be the
greatest of all impediments to scientific
literacy.
Stephen Jay Gould
The Personal Probability Interpretation
Personal probability of an event = the degree
to which a given individual believes the event
will happen.
Sometimes subjective probability used
because the degree of belief may be different
for each individual.
Restrictions on personal probabilities:
• Must fall between 0 and 1 (or between 0 and 100%).
• Must be coherent.
Probability Definitions and
Relationships
Sample space: All the possible outcomes that can
occur.
Simple event: one outcome in the sample space; a
possible outcome of a random circumstance.
Event: a collection of one or more simple events in
the sample space; often written as
A, B, C, and so on.
Assigning Probabilities
• A probability is a value between 0 and 1 and
is written either as a fraction or as a
proportion.
• A probability simply is a number between 0
and 1 that is assigned to a possible outcome
of a random circumstance.
• For the complete set of distinct possible
outcomes of a random circumstance, the total
of the assigned probabilities must equal 1.
Classical Approach
• A mathematical index of the relative
frequency of likelihood of the occurrence
of a specific event.
– Based on games of chance
– The specific conditions of the game are
known.
Determining the probability of an
Outcome (Classical)
A Simple Lottery
Choose a three-digit number between 000 and 999.
Player wins if his or her three-digit number is chosen.
Suppose the 1000 possible 3-digit numbers (000, 001, 002,
999) are equally likely.
In long run, a player should win about 1 out of 1000
times. Probability = 0.001 of winning.
This does not mean a player will win exactly once in every
thousand plays.
Example: Probability of Simple Events
Random Circumstance:
A three-digit winning lottery number is selected.
Sample Space: {000,001,002,003, . . . ,997,998,999}.
There are 1000 simple events.
Probabilities for Simple Event: Probability any specific
three-digit number is a winner is 1/1000.
Assume all three-digit numbers are equally likely.
Event A = last digit is a 9 = {009,019, . . . ,999}.
Since one out of ten numbers in set, P(A) = 1/10.
Event B = three digits are all the same
= {000, 111, 222, 333, 444, 555, 666, 777, 888, 999}.
Since event B contains 10 events, P(B) = 10/1000 = 1/100.
Estimating Probabilities from Observed
Categorical Data - Empirical Approach
Assuming data are representative, the
probability of a particular outcome is
estimated to be the relative frequency
(proportion) with which that outcome
was observed.
Methods of sampling
• Simple random selection
– Every member of the population has an equal
chance of being selected.
• Systematic
– Every Xth person.
• Stratified
– Random sampling by subgroup.
• Why?
Determining the probability of an
Outcome – Empirical Approach
Observe the Relative Frequency of
random circumstances
The Probability of Lost Luggage
“1 in 176 passengers on U.S. airline carriers will
temporarily lose their luggage.”
This number is based on data collected over the
long run. So the probability that a randomly
selected passenger on a U.S. carrier will
temporarily lose luggage is 1/176 or about 0.006.
Proportions and Percentages as
Probabilities
• The proportion of passengers who lose their
luggage is 1/176 or about 0.006 (6 out of 1000).
• About 0.6% of passengers lose their luggage.
• The probability that a randomly selected
passenger will lose his/her luggage is about 0.006.
• The probability that you will lose your luggage
is about 0.006.
Last statement is not exactly correct – your probability
depends on other factors (how late you arrive at the
airport, etc.).
Example: Probability of Male
versus Female Births
Long-run relative frequency of males born in the United
States is about 0.512 (512 boys born per 1000 births)
Table provides results of simulation: the proportion is far from
.512 over the first few weeks but in the long run settles down
around .512.
Nightlights and Myopia
Assuming these data are representative of a larger
population, what is the approximate probability that
someone from that population who sleeps with a
nightlight in early childhood will develop some degree of
myopia?
Note: 72 + 7 = 79 of the 232 nightlight users developed some
degree of myopia. So we estimate the probability to be
79/232 = 0.34.
Complementary Events
One event is the complement of another event if the two
events do not contain any of the same simple events and
together they cover the entire sample space.
Notation: AC represents the complement of A.
Note: P(A) + P(AC) = 1
Example:A Simple Lottery (cont)
A = player buying single ticket wins
AC = player does not win
P(A) = 1/1000 so P(AC) = 999/1000
Mutually Exclusive Events
Two events are mutually exclusive if they do not
contain any of the same simple events (outcomes).
Example; A Simple Lottery
A = all three digits are the same.
B = the first and last digits are
different
The events A and B are mutually
exclusive.
Independent and Dependent Events
• Two events are independent of each other
if knowing that one will occur (or has
occurred) does not change the probability
that the other occurs.
• Two events are dependent if knowing that
one will occur (or has occurred) changes
the probability that the other occurs.
Example Independent Events
•Customers put business card in restaurant
glass bowl.
•Drawing held once a week for free lunch.
•You and Vanessa put a card in two consecutive
wks.
Event A = You win in week 1.
Event B = Vanessa wins in week 2
• Events A and B refer to to different random
circumstances and are independent.
Example: Dependent Events
Event A = Alicia is selected to answer Question 1.
Event B = Alicia is selected to answer Question 2.
Events A and B refer to different random circumstances,
but are A and B independent events?
• P(A) = 1/50.
• If event A occurs, her name is no longer in the bag; P(B) = 0.
• If event A does not occur, there are 49 names in the
bag (including Alicia’s name), so P(B) = 1/49.
Knowing whether A occurred changes P(B). Thus, the
events A and B are not independent.
Joint and Marginal Probabilities
• These probabilities refer to the proportion of an
event as a fraction of the total.
Unions and intersections
• P{AB}  P{A} + P{B} because A and B
do overlap.
• P{AB} = P{A} + P{B} - P{AB}.
• AB is the intersection of A and B; it
includes everything that is in both A and B,
and is counted twice if we add P{A} and
P{B}.
Conditional Probability
• Consider two events A and B.
• What is the probability of A, given the
information that B occurred? P(A | B) = ?
• Example:
– What is the probability that a women is
married given that she is 18 - 29 years old?
Probability Problems
P(Married | 18-29) = 7842/ 22,512
Conditional probability and
independence
• If we know that one event has occurred it may change
our view of the probability of another event. Let
– A = {rain today}, B = {rain tomorrow}, C = {rain in 90 days time}
• It is likely that knowledge that A has occurred will change
your view of the probability that B will occur, but not of
the probability that C will occur.
• We write P(B|A) P(B), P(C|A) = P(C). P(B|A) denotes
the conditional probability of B, given A.
• We say that A and C are independent, but A and B are
not.
• Note that for independent events P(AC) = P(A)P(C).
Conditional probability - tornado
forecasting
• Consider the classic data set on the next Slide
consisting of forecasts and observations of
tornados (Finley, 1884).
• Let
– F = {Tornado forecast}
– T = {Tornado observed}
• Use the frequencies in the table to estimate
probabilities – it’s a large sample, so estimates
should not be too bad.
Forecasts of tornados
Tornado
observed
No T
observed
Total
Tornado No T
Total
Forecast forecast
28
23
51
72
2680
2752
100
2703
2803
Conditional probability - tornado
forecasting
• P(T)
= 51/2803 = 0.0182
• P(TF) = 28/2803
• P(T|F) = 28/100 = 0.2800
• P(T|Fc) = 23/2703 = 0.0085
– Knowledge of the forecast changes P(T). F and T are
not independent.
• P(F|T) = 28/51 = 0.5490
– P(T|F), P(F|T) are often confused but are different
quantities, and can take very different values.
Continuous and discrete random
variables
• A continuous random variable is one which can
(in theory) take any value in some range, for
example crop yield, maximum temperature.
• A discrete variable has a countable set of
values. They may be
– counts, such as numbers of accidents
– categories, such as much above average, above
average, near average, below average, much below
average
– binary variables, such as dropout/no dropout
Probability distributions
• If we measure a random variable many times,
we can build up a distribution of the values it can
take.
• Imagine an underlying distribution of values
which we would get if it was possible to take
more and more measurements under the same
conditions.
• This gives the probability distribution for the
variable.
Continuous probability distributions
• Because continuous random variables can take
all values in a range, it is not possible to assign
probabilities to individual values.
• Instead we have a continuous curve, called a
probability density function, which allows us to
calculate the probability a value within any
interval.
• This probability is calculated as the area under
the curve between the values of interest. The
total area under the curve must equal 1.
Normal (Gaussian) distributions
• Normal (also known as Gaussian) distributions
are by far the most commonly used family of
continuous distributions.
• They are ‘bell-shaped’ –and are indexed by two
parameters:
– The mean m – the distribution is symmetric about this
value
– The standard deviation s – this determines the
spread of the distribution. Roughly 2/3 of the
distribution lies within 1 standard deviation of the
mean, and 95% within 2 standard deviations.
The probability of continuous
variables
• IQ test
– Mean = 100 and sd = 15
• What is the probability of randomly
selecting an individual with a test score of
130 or greater?
– P(X ≤ 95)?
– P(X ≥ 112)?
– P(X ≤ 95 or X ≥ 112)?
The probability of continuous
variables (cont.)
• What is the probability of randomly
selecting three people with a test score
greater than 112?
– Remember the multiplication rule for
independent events.