Download Introduction to Probability I

Introduction to Probability I
Eleisa Heron
Neuropsychiatric Genetics Research Group
Trinity College Dublin
15/10/08
Introduction
“Probability and statistics used to be
married; then they separated; then they got
divorced; now they hardly ever see each
other.”
D. Williams, Weighing the Odds (2001)
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Introduction
Statistical Analysis
Descriptive Statistics
Statistical Inference
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Descriptive Statistics: organising, presenting and summarising data
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Statistical inference: observe a sample from a population and want
to infer something about the population - need probability!
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Outline
• 2 lectures:
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Origins of probability, frequentist, Bayesian
Definitions
Probability Rules
Mutually Exclusive Events
Non-Mutually Exclusive Events
Conditional Probability
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Bayes’ Theorem
Independent Events
Marginal Probability
Odds
Fun Problems!
References
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Origins of Probability
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We all have an intuitive idea of what probability is
Can think of probability as a mathematical expression of the relationship
between a particular outcome of an experiment and the total number of
possible outcomes:
toss a coin:
roll a dice:
(assumptions?)
•
Modern probability – gambling
17th Century – mathematicians Blaise Pascal and Pierre Fermat worked out the
theory of probabilities in response to a gambling problem
Many others along the way …
•
Foundations of probability 1930’s Russian mathematician A.N. Kolmogorov
(measure theory, branch of pure mathematics)
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Probability Interpretations – Frequentist vs Bayesian
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Most use probability unthinkingly in a “frequentist way”
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Frequentist - probability of an event is given by the fraction of times the event
occurs on average over a large number of trials/experiments
– Approximation to true probability
– Improves as number of trials increases
– Assumption that trial can be repeated many times
(always valid? – winning an election, passing a test ??)
•
Bayesian (subjective) probability - reflects a person’s opinion about how
likely an event is to occur, it represents the person’s strength of belief
– Prior beliefs are updated using observed data
– Valid in situations where long run relative frequency is not applicable
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Probability Interpretations – Frequentist vs Bayesian
What’s the correct approach?
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Long run relative frequency idea of probability: intuitive motivation
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Long run relative frequency can’t be made mathematically rigorous
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Measure theory takes care of this problem
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Once the rules of probability are defined – individual’s choice on
interpretation as frequentist or Bayesian
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Definitions
Trial
any process, which when repeated, generates a set of
results or observations (trial = experiment)
eg. Tossing a coin, rolling a dice
Outcome
the result of carrying out a trial
eg. A head, a 4
Event
a set consisting of one or more of the possible
outcomes of a trial
eg. The event getting a head, the event getting a 4 or a 6
Sample Space
the set of all possible outcomes of a trial
eg. Toss a coin – head and tail
Throw a dice – 1, 2, 3, 4, 5, and 6
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Probability Rules
1. For an event
2.
3.
is defined to be the event
not occurring
If an event
cannot occur:
if an event
will definitely occur:
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Mutually Exclusive Events – Addition Rule
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Two or more events are said to be mutually exclusive if the events cannot
occur simultaneously
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For mutually exclusive events
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Sum of probabilities of all mutually exclusive events is 1
OR means ADD for mutually exclusive events
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Mutually Exclusive Events – Example
Example: A card is dealt from a pack of well-shuffled cards.
What is the probability that the card is either a heart or a spade?
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Hearts
Clubs
Diamonds
Spades
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Example: Toss a Single Coin
Getting a head
A head not occurring
Getting 5 heads
Getting a head or a tail
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Example: Roll a Dice
Getting a 1
Getting a 4
Getting a number greater than 1
OR
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Non-Mutually Exclusive Events
•
Mutually Exclusive?
– The events getting a head on tossing a coin and getting a tail?
– The events getting a six and getting a three when throwing a dice?
– The events getting an ace and getting a diamond when we pick a card at random?
•
If two events
and
are not mutually exclusive,
can both happen together:
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Example: Group of 10 people, 5 taking drug A, 3 people taking drug B, 2
people taking drug A and drug B and 4 people taking no drug,
choose a person at random
– Probability the person takes drug A or drug B?
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Non-Mutually Exclusive Events
Drug A
Drug B
3
2
1
4
Example: Group of 10 people, 5 taking drug A, 3 people taking drug B, 2 people
taking drug A and drug B and 4 people taking no drug, choose a person
at random
– Probability the person takes drug A or drug B?
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Conditional Probability
•
Two events:
•
Example: Group of 10 people, 5 taking drug A, 3 people taking drug B, 2
people taking drug A and drug B and 4 people taking no drug,
choose a person at random
and ,
occurring, given that event
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is the conditional probability of event
occurred
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Conditional Probability Fallacy
•
Explore difference between
•
Example:
and
– For a particular disease, 1% of the population suffers from the disease
– Disease is curable if detected early
– Screening test for the disease is 99% accurate (99% sensitivity and
specificity)
Want to explore the idea of screening people for the disease, but
want to think about the distress that would be caused to someone who
doesn’t have the disease but is incorrectly found to have the disease
by the screening test
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Conditional Probability Fallacy Cont’d
•
Example:
– For a particular disease, 1% of the population suffers from the disease
– Screening test for the disease is 99% accurate (99% sensitivity and
specificity)
(sensitivity)
(specificity)
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Conditional Probability Fallacy Cont’d
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Have
•
Can work out:
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Conditional Probability Fallacy Cont’d
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What is the probability of having the disease GIVEN that the test gave
a positive result?
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Surprising result?
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50% of positives would be false positives – unacceptable
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Difference between the probability of testing positive given that the
person has the disease (0.99) and the probability of having the
disease given that the person tests positive (0.5)
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