Introduction to Probability II - Trinity College, Dublin · Introduction to Probability II ... is the probability of the event in the second group ... • Introduction to Probability

Introduction to Probability II

Eleisa Heron

Neuropsychiatric Genetics Research Group

Trinity College Dublin

22/10/08

22/10/08 2Probability II E. Heron

Review – Probability I

– Origins of probability – 17th C. gambling, frequentist and Bayesian interpretations

– Definitions – trial, outcome, event, sample space

– Probability Rules – [0,1], impossible and certain events

– Mutually/Non-Mutually Exclusive Events – OR means add rule (ME) Venn diagrams

– Conditional Probability – conditional probability fallacy


Bayes’ Theorem

• Conditional probability for event given that event has occurred

• Using this we can say

• Giving Bayes’ theorem


Bayes’ Theorem and Bayesian Inference

• In Bayesian inference, given data and parameters of a model, we want to estimate the parameters

• Sometimes, Bayes’ theorem is written in the following way


Bayes’ Theorem Example

• Court Setting

Suppose a juror wants to combine DNA evidence that is presented to him/her together with their own prior beliefs in order to decide if defendant is guilty or not

Evidence: DNA that matches the defendant’s DNA was found at the crime scene

• is the event the defendant is guilty

• is the event the defendant is not guilty• i is the event that the defendant’s DNA matches the DNA found at the crime

scene

• Interested in

• Probability of being guilty before any evidence is presented

• Probability of the event


Bayes’ Theorem Example

• From Bayes’ theorem we have

• Suppose a male committed the crime in a town with a male population of 20,000

• Juror uses this for his prior

• This gives


Heads and tails – 0’s and 1’s

10110011001111111110 11010110000111111010 00100111000010000001 00101111001001000011 10110100110000110010

10100101001010010110

01010110110101011010

00100101010010110101

10101011001010010110

01001101010010110010

• Which sequence have I made up and which is a true simulation?


Boys and Girls

• Five children in a family,

assume probability of a boy (B) = probability of a girl (G) = 0.5

GGGGG

BBGBG

BBBBB

Which has higher probability?


Independent Events

• Two events and are independent if

• Toss a coin and get a head

Toss the coin again

Probability of getting a head again is independent of the first toss and the fact that we got a head


Independent Events

• For independent events and then

• Toss two coins

AND means MULTIPLY for independent events


Independent Events

• Can mutually exclusive events be independent?

• Assuming A and B are non-trivial events

Two events A and B

Mutually exclusive Not mutually exclusive

Independent Dependent


Marginal Probability

• . the marginal probability of an event is the unconditional probability of the event

• The probability of the event regardless of what other events occurred

• Disease screening example

Positive Negative

Disease

Well

P(Dis, Pos)

P(Well, Pos) P(Well, Neg)

P(Positive)

P(Dis, Neg)

P(Negative)

P(Well)

P(Disease)


Odds

• Odds of an event are defined as the ratio of the probability that the event will occur to the probability that the event will not occur

• To convert from odds to probability

• Probability more intuitive, but odds used in gambling and in analysis of binary outcome variables (logistic regression) for example


Odds Cont’d

• Odds lie between and , since probability lies between and


Odds Ratio

• Odds ratio (OR) is a measure of effect size

• OR defined as the ratio of the odds of an event occurring in one group to the odds of the event occurring in another group

is the probability of the event in the first group

is the probability of the event in the second group

OR =1: the condition or event is equally likely in both groups

OR >1: the condition or event is more likely in the first group

OR <1: the condition or event is less likely in the first group

• Again, just like odds, OR’s lie between and


Pink and Blue Cards

• Pick a card– What is the probability card is blue on both sides?

• Pick a card, card is blue on one side– What is the probability card is blue on the other side?


Pink and Blue Cards Cont’d

Pink

Pink

Pink

Blue

BlueBlue

1/3

1/3

1/3

Card Picked

Pink Pink

Pink Pink

Pink Blue

Blue

Blue

Blue Blue

Blue

Pink

1/2

1/2

1/2

1/2

1/2

1/2

First Second


Pink and Blue Cards Cont’d

• is the event that the first card is blue and is the event that the second card is blue


Goats and Car – Monty Hall Problem

• Game Show

– 3 doors, a car behind one and a goat behind each of the others

– Contestant picks one of the 3 doors

– Game show host (who knows which door conceals the car) opens one of the remaining two doors to reveal a goat

– Contestant offered the chance to swap the door they originally chose for the remaining door

Should the contestant swap?


Assume Door 1 is chosen

Goats and Car – Monty Hall Problem Cont’d

Door 11/3

1/3

1/3

Car location

Door 2

Door 3

Don’t Swap Swap

Car

Car

Car

Car

Goat

Goat

Goat

Goat

Prob. Total

1/6

1/6

1/3

1/3

1/2

1/2

Host Opens

Door 2

Door 2

Door 3

Door 31

1



Assume contestant chooses door 1 and the presenter opens door 3

is the event that the presenter opens door 3

is the event that the car is behind door 1, similarly and




References

• Most introductory statistics books have a probability section at the start

• Statistics for Technology by C. Chatfield

Basic introductory probability

• Introduction to Probability by Charles M. Grinstead, J. Laurie Snell (1997) More in depth

• Weighing the Odds: A Course in Probability and Statistics by D. Williams (2001)

More advanced, looking at both frequentist and Bayesian approaches

Documents

Introduction to Probability II - Trinity College, Dublin · Introduction to Probability II ... is the probability of the event in the second group ... • Introduction to Probability