Review 

Random Phenomena 

We call a phenomenon "random" if individual outcomes are uncertain but there is, more the less, a regular distribution of relative frequencies in a large numbers or repetitions. By 'regular' we mean that the distribution is stable or unchanging. 

Random phenomena are not chaotic (because they are not regular) 

Probability = long run relative frequency 

• e.g. flips of dice and coins - note how relative frequency stabilizes
• problems - verifiability

Ultimately we want to study the origin of lists of numbers — those that come from probability models, with a view to understand the process by which data in the list were generated. 

Mathematics of Probability 

1. Experiment/random phenomena 
2. Sample space - set 5 of all possible outcomes e.g. flip a coin 
3. Towards a consistent theory of assigning probabilities

• a subset of the sample space.
• e.g. flip a coin 4 times

• elementary events are the building blocks of events
• elementary event are disjoint - both can't happen at the same time

Assigning Probabilities to Events 

Rule 1 Any probability P(A) is a number between 0 and 1

0 # P(A) # 1 

Rule 2 The collection S of all possible outcomes (elementary events) has probability 1. P(S) = 1 

e.g. Assign probability p to flip of a coin (e.g. P = .5 if the coin is fair).

If A is any event consisting of (disjoint) outcomes (= elementary events) then Prob. of A= sum of probabilities of individual outcomes 

if elementary events = outcomes are equally probable then: 

e.g. roll die 

Venn Diagrams {event, disjoint events) 

Rule 3 If events A and B are disjoint the:

P(A c B) = P(A or B) = P(B or A) = P(A) + P(B) 

Complement of Event

A and A^c are disjoint sets which partition the sample space S into 2 parts. 

Rule 4 Complement 

For any event A, the probability that A does not occur is:

P(A^c) = 1 - P(A) 

Dependent and Independence 

{A, A^c} are not independent - if one happens the other can’t 

e.g. 2 flips of a fair coin - flip a coin/check hair colour 

Rule 5 (independent events: multiplication) 

If events A and B are independent then:

P(A Ù B) = P(A and B) = P(A) × P(B) 

Rule 3 (extended)

P(A È B È C) = P(A) + P(B) + P(C) 

If A and B and C are all disjoint. 

General Rule 

P(A or B) = P(A È B) = P(A) + P(B) - P(A Ç B) 

A and B don’t need to be disjoint. 

Contingency Table 

Conditional Probability P(A|B)…What does this mean? 

Joint Probability P(A Ç B)… A and B not necessarily independent

e.g. … police force  

Total Probability:  

let A₁, A₂, …A_m be a collection of disjoint events so that A_{1 È} A₂ È A₃ È …È A_m = S 

Then P(B) = P(B Ç A₁) + P(B Ç A₂) + … + P(B Ç A_m) 

Conditional Probability 

Venn diagram shows this is just a restriction of S. 

General Multiplication Rule for Intersections 

P(A Ù B) = Prob. (A and B) = P(A) P(B|A)

= P(B) P(A|B) 

Reverse Conditional Probability (Bayes Problem) 

Usually given P(A), P(B), P(C|A), P(C|B) 

Asked to find P(A|C) or P(B|C) 

Steps: get P(C) = P(C|A) × P(A) + P(C|B) × P(B)

P(A Ù C) + P(B Ù C)

Random Variables 

A random variable is a variable whose value is the numerical outcome of a random phenomena. 

Discrete Random Variables 

 Continuos Random Variable 

• how to get an outcome
• probability density (area = 1)
• e.g. uniform, normal
• probability of event = area under density and above interval (event = interval or set of intervals) 

List of numbers = realization of random variables (maybe all with the same distribution) 

MEAN 

m _x = å p_ix_i (derive from Law of Large Numbers)

= long run average 

Law of Averages - how long does it take? Law of Small Numbers 

m _a+bx = a + bm _x

m _x+y = m _x + m _y  

Variances of Random Variables 

if X and Y are independent

Random variables related to counts and proportion.

Binomial Setting

Binomial B(n,p)

å probability of any one way

ä

# of ways

Use of tables 

Approximation of Binomial Probabilities by normal 

n large enough

p close enough to .5 

Can also work for events such as P(x k) or P(X > k) 

Proportions 

X is B(n,p)

for

Sample Statistics — Sample mean

Central Limit Theorem distribution of  

is an estimator, is a point estimate 

is unbiased for  

Illustrated of CLT for various population