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Probability Certainty

Overview

- History of probability theory
- Everything you need to know about probability on

one! - Some basic probability theory
- Calculating simple probabilities
- Combining mutually-exclusive probabilities
- Combining independent probabilities
- More complex probabilities
- Calculating conditional probabilities
- Bayes rule, and why we should care about it
- A devious test case The notorious Lets Make A

Deal! problem

Review

- Last time we made three main points
- Information is related to elimination of

redundancy in any dataset - Information is related to purpose or goals
- Information in the real world is always uncertain

or probabilistic

Who should you care

- Probability theory comes into play in four main

ways in this course - i.) As the explanation behind distributional

regularities (normalcy) upon which we will rely

very heavily to build some useful statistical

tools - ii.) In understanding certain systematic errors

people make in reasoning about diagnosis - iii.) In understanding how base rates of a

disease or state can impact on our diagnosis of

that disease or state - iv.) In understanding how to decide where to put

cut-off points for diagnosing a person as

belonging to a specific diagnostic category - Generally, probability theory underlies much of

the reasoning in psychometrics

History of probability theory

- Compress all of human history (350K years) in one

24-hour day - The first recorded general problem representation

(geometry, invented by Thales of Miletus about

450 B.C.) would have appeared only 9 minutes and

30 seconds ago - The first systematic large-scale collection of

empirical facts (Tycho Brahes collection of

astronomical observations) would have appeared

only a minute and a half ago - The first mathematical equation which was able to

predict an empirical phenomena (Newtons 1697

equation for planetary motion) would have

appeared only one minute and twelve seconds ago - Probability theory appeared between 1654 (a

minute and a half ago) and 1843 (34 seconds ago).

History of probability theory

- The emergence of elementary probability theory in

the 1650s met with enormous resistance and lack

of comprehension when it was first introduced,

despite its formal character, its utility, and

(what we now recognize as) its simplicity. - The difficult points were philosophical rather

than mathematical (Margolis, H. (1993). Paradigms

and barriers How habits of mind govern

scientific beliefs. Chicago University Press.)

History of probability theory

- In particular, even the greatest geniuses of the

time had difficulty wrapping their minds around

two notions - i.) That one could (and, indeed, had to) count

possibilities that had never actually existed and

never would - ii.) That order could be obtained from

randomness.

Everything you need to know about probability for

this class.

- Basic principle The probability of any

particular event is equal to the ratio of the

number of ways the event can happen over the

number of ways the event can fail to happen the

number of ways it can happen - To combine probabilities of independent events

(unrelated ands), multiply the odds of each

event. - To combine probabilities of mutually exclusive

events (either/ors), add the odds of each

event. - Bayes Rule When one probability is conditional

on another - P(AB) P(A and B) / P(B)

Basic probability theory Example 1

- A boring standard example

How likely is it that we will throw a 6 with one

dice

- Basic principle The probability of any

particular event is equal to the ratio of the

number of ways the event can happen over the

number of ways anything can happen ( the number

of ways the event can fail to happen the number

of ways it can happen).

Basic probability theory Example 2

How likely is it that we will throw a 7 with two

dice

Basic probability theory Example 3

We roll the die 4 times, and never get a seven.

What is the probability that well get on the

5th roll

- Independent events are events that dont effect

each others probability. Since the every roll is

independent of every other, the odds are still

1/6.

Basic probability theory Example 4

We roll the die twice and get a seven both times.

What is the probability of that

- Are the two events dependent or independent

Basic probability theory Example 4

We roll the die twice and get a seven both times.

What is the probability of that

Basic probability theory Example 5

We roll the die. What are the odds are getting

either a 7 or a 2

- Now the events are mutually exclusive if one

happens, the other cannot. To combine

probabilities of mutually exclusive events, add

them together.

Basic probability theory Example 6.1

We roll the die twice. What are the odds that we

get at least one 7 from the two rolls

- Can we just add the probabilities of getting a 7

on each roll

Basic probability theory Example 6.2

We roll the die twice. What are the odds that we

get at least one 7 from the two rolls

- Can we just multiply the probabilities of

getting a 7 on each roll

Basic probability theory Example 6.3

We roll the die twice. What are the odds that we

get at least one 7 from the two rolls

- We can turn part of the problem into a problem

of mutual exclusivity by asking what are the

odds of there being exactly one seven out of two

rolls

- One way is to roll 7 first, but not second
- - The odds of this are 1/6 5/6 (independent

events) 0.138 - Similarly, the odds of rolling 7 second are 5/6

1/6 (independent events) 0.138 - - Since these two outcomes are mutually

exclusive, we can add them to get 0.138 0.138

0.277

Basic probability theory Example 6.4

We roll the die twice. What are the odds that we

get at least one 7 from the two rolls

- Now we need to know what are the odds of there

being two sevens out of two rolls

- We already know its 1/36 0.03
- So the odds that we get at least one 7 is the

odds of two 7s the odds of one 7 (mutually

exclusive events, so we sum) 0.03 0.27 0.3

Basic probability theory The generalization

- Does anyone know how to generalize this

calculation, so we can easily calculate the odds

of an event of probability p happening r times

out of n tries, for any values of p,r, and n - What would we get if we generalized it across all

values of r,n with p 0.5 (coin flips)

Basic probability theory Example 6.5

- We draw a card from a full deck. What is the

probability it is either red or a face card - To or events A and B, we need the addition

rule - P(A) P(B) P(A and B)
- If we dont subtract out the cards to which both

conditions apply, their probability is weighted

double

Basic probability theory Example 6.5

- We draw a card from a full deck. What is the

probability it is either red or a face card - P(A) P(B) P(A and B)
- P(red card)
- P(face card
- P(red face card)

Basic probability theory Example 6.6

- We draw a card What is the probability it is

lower than a queen and black - Here it is easier to subtract out what doesnt

apply than to add in what does

Basic probability theory Conditional probability

- What if a dice is biased How can we deal with

relevant prior probabilities - P(Roll a six with cheatin dice) P(Roll a six

with a fair dice) - Why should we care about such an arcane example
- Because real life almost always uses biased dice
- Eg. the conditional probability of being

schizophrenic, given that a person has an

appointment with a doctor who specializes in

schizophrenia, is quite different from the

unconditional probability that a person has

schizophrenia (the base rate)

Conditional probability

- Conditional probability arises when one

probability P(A) depends on another probability

P(B) which is defined over the same population

of events - We say that we want to know P(A) given P(B)

notationally, P(AB) - We can say that the word given defines a subset

of the population of events namely that subset

that depends on P(B)

Conditional probability

- More formally, what P(AB) says is Pick out the

events to which both P(A) and P(B) apply, and

consider them as part of the subset of events to

which only P(B) applies hence - P(AB) P(A and B) / P(B)
- P(A,B) / P(B) A notational change only

Example Probability of cancer given that youre

female

- For example, what P(Cancer F) says is Pick out

the people to whom both P(Cancer) and P(F) apply

( woman with cancer), and consider them as part

of the subset of events to which only P(F)

applies (e.g. woman) hence - P(Cancer B) P(Cancer and F) / P(F)

Have skin cancer

Women

Woman with skin cancer

Example Probability of cancer given that youre

female

- All we are doing with P(Cancer F) P(Cancer

and F) / P(F) is selecting out the same subset of

the event population in both the first and second

parts of the conditional in this case, only

women

Have skin cancer

Women

Woman with skin cancer

Example Probability of cancer given that youre

female

- In other words, we are merely asking What

proportion of woman have cancer P(Cancer F)

P(Cancer and F) / P(F)

Have skin cancer

Women

Woman with skin cancer

Conditional probability

- What we are doing with P(AB) P(A and B) / P(B)

is selecting out the same subset of the event

population in both the numerator and the

denominator - The reason it is so important to do this is

because in some cases P(A) in some relevant

subset of an event population is very different

from P(A) in the whole event population - For example, P(Have healthcare Canadian) is

very different from P(Have healthcare) - Everyone in Canada has healthcare P(Have

healthcare Canadian) 1 - Most people in the world do not have healthcare

P(Have healthcare)

A generalization Bayes rule

- P(AB) P(BA) P(A) / P(B)
- Some people find Bayes rule helpful, since in

some cases it can clarify the problem being

considered - Some people find it more confusing than helpful,
- The important point to understand is that Bayes

rule is just a re-statement of the definition of

conditional probability, not a new finding - Note Multivariate version of Bayes rule are

defined we dont use them in this class, in

which we only consider two-possibility problems.

A generalization Bayes rule

- P(AB) P(BA) P(A) / P(B)
- Note that all this says is P(A and B) P(BA)

P(A) because we already saw that P(AB) P(A

and B) / P(B) - So The probability of having being a Canadian

and having health care is the same as the

probability of having health care given that you

are Canadian (1) X the probability of being

Canadian (0.004) - If only females had health care, then the

probability of being Canadian and having health

care would be half as much the probability of

having health care given that you are Canadian

(0.5) X the probability of being Canadian (0.004)

A generalization Bayes rule

- P(AB) P(BA) P(A) / P(B)
- Note that all this says is P(A and B) P(BA)

P(A) - Another intuitive example The probability of

winning the lottery and buying a ticket is

dependent both on the probability of winning the

lottery when given that you have a ticket and

on the probability that you buy a ticket - A person with a low probability of buying a

lottery ticket has a lower probability of holding

a winning ticket than a person that buys more

tickets even though P(WinningTicket)- the odds

of any particular ticket being a winning ticket-

is the same for both people

A generalization Bayes rule

P(AB) P(BA) P(A) / P(B) Proof By

definition, (1.) P(AB) P(A,B) / P(B) (2.)

P(BA) P(A,B) / P(A) (3.) P(AB) P(B) P(A,B)

Multiply (1.) by P(A) (4.) P(BA) P(A) P(A,B)

Multiply (2.) by P(A) (5.) P(AB) P(B)

P(BA) P(A) Substitute (4.) in (3.) (6.) P(AB)

P(BA) P(A) / P(B) Divide by P(B)

One more example

- P(Female Second row)
- The relevant numbers are something like this
- in class 35
- Females in class 30
- Number in row 2 7
- Males in row 2 2
- Female in row 2 5

Another example

- The relevant numbers are something like this
- in class 35
- Females in class 30
- Number in row 2 7
- Males in row 2 2
- Females in row 2 5
- P (Female and Row 2) 5/35
- P(R2) 7/35
- P(F2) P(F R2) / P(R2)
- 5/35 / 7/35
- 5/7

The failings of Harvard Medical School

- A particular disorder has a base rate occurrence

of 1/1000 people. A test to detect this disease

has a false positive rate of 5. Assume that the

test diagnoses correctly every person who has the

disease. What is the chance that a randomly

selected person with a positive result actually

has the disease Take a guess.

The failings of Harvard Medical School

- A particular disorder has a base rate occurrence

of 1/1000 people. A test to detect this disease

has a false positive rate of 5. Assume that the

test diagnoses correctly every person who has the

disease. What is the chance that a randomly

selected person with a positive result actually

has the disease

- Conditional probability P(AB) P(A,B) / P(B)

P(A and B)/P(B) - Let A Has the disorder and B Has a positive

test result - In 10,000 people, P(A and B) See the answer

in class - P(B) See the answer in class
- Chance that a randomly selected person with a

positive result actually has the disease See

the answer in class

Lets try Bayes rule

P(AB) P(BA) P(A) / P(B) Let P(A)

Probability of disease 0.001 P(B)

Probability of positive test 0.05 0.001

0.0501 P(BA) Probability of positive

test given disease 1 Then P(AB) P(BA)

P(A) / P(B)

The Notorious 3-Curtain (Lets Make A Deal)

Problem!

- Three curtains hide prizes. One is good. Two are

not. - You choose a curtain.
- The MC opens another curtain. Its not good.
- He gives you the chance to stay with our first

choice, or switch to the remaining unopened

curtain. - Should you stay or switch, or does it matter

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