Probability I

Sample Spaces• Sample space:• The set of possible outcomes for a question or experiment

Experiment: Flip a two sided coin

Question: Did the North Korean government hack Sony?

Sample Spaces• The number of outcomes in a sample space may

be enormous

Question: How many 5-card poker hands are possible?

Question: How unique human DNA “profiles” are there?

Sample Spaces• Outcomes can be continuous or discrete

Discrete: Did he do it?

Continuous: What is the mass of scheduled drugs seized?

Sample Spaces• An event:• Subset of a sample space

E

Sample Spaces• The complement to the event:• Everything not in the event

EE’

Sample Spaces

• A simple event is an event containing a single outcome.

• A compound event consists of more than one outcome.

• When the experiment is performed, if the outcome that occurs is in event E then we say E occurs.

Sample Spaces• Example: Toss a coin:– Ω = {H,T}. – The event “head” is {H} and is a simple event.

• Example: Toss a coin twice – There are several possible sample spaces: – Ω = {0,1,2} : the number of heads appearing. – Ω = {HH,HT,TH,TT} : 1st toss, 2nd toss.

• The event {HT,TH} corresponds to getting exactly one head. It is a compound event.

Sample Spaces• Example: Petraco et al. has built a database of common

constituents in dust from varied and widespread locations:

539 categories of (semi)common components

e.g.: "Human Hair Natural " " Head" "C Blk"

• Venn diagram: A pictorial representation of combinations of sets making use of circles and rectangles.

Some More Set Theory Language

• Empty set: The set containing no outcomes. • The null set or { }.

• Union: A B occurs if A occurs, B occurs or both A and B occur.

A BA B

Some More Set Theory Language

• Intersection: A B occurs if both A and B occur.

• Disjoint: A and B are disjoint or mutually exclusive if they have no outcomes in common, i.e. if A B = .

A BA B

A B

Kolmogorov Axioms of Probability• Axiom 1: For any event A, Pr(A) ≥ 0

• Axiom 2: Pr(Ω) = 1

• Axiom 3: For a collection of mutually exclusive events, A1, A2, …, An

• Everything else in probability theory can be deduced starting with these axioms

Kolmogorov Axioms of Probability

• Important consequences:• A probability function assigns a probability to any event A such

that:

• A partition of the sample space means:

In words: The Ai’s chop up the sample space into non-overlapping (i.e. mutually exclusive) pieces.

Kolmogorov Axioms of Probability

• Important consequences:• Probability of a complement

• Probability of nothing in the sample space

Kolmogorov Axioms of Probability

• Important consequences:• Probability of a union of non-disjoint events

In words: The probability of A or B is the probability of A plus the probability of B minus the probability of A and B

Don’t count the probabilities of A and B twice if there is overlap between the events

• Three or more events are done similarly

Handy: DeMorgan’s Laws

DeMorgan Law 1

DeMorgan Law 2

Kolmogorov Axioms of Probability

Example

Example

Sally got shot by a purp.

Let A = Joe shot Sally. Pr(A) = 0.3Let B = Bill shot Sally. Pr(B) = 0.5

• Draw a Venn diagram for this scenario• Compute• Compute• Compute

Counting Formulas

• When various outcomes of an experiment are equally likely computing probabilities reduces to a counting problem

• Product rule for ordered k-tuples:• k-tuple = (item1, item2, …, itemk)• 2-tuple = a pair

• item1 has n1 possibilities, item2 has n2 possibilities, …

• Total number of ways to select k items = n1n2…nk

Example

In a computer integers are represented as a pattern of 32 ones and zeros. How many possible “bit patterns” are there to represent integers in the scheme?

Example

How many different 7-place license plates are possible if the first 3 are to be occupied by letters and the final 4 by numbers?

How would this be different if repetition among letters and numbers is not allowed?

Counting Formulas• How many ways are there to select r distinct items from a

group of n distinct items?

• Permutations: If the order of selection is important

• Combinations: If the order of selection is irrelevant

Example

Prof. Shenkin has 10 books that he wants to put on the shelf. Of these 4 are about probability, 3 are about algebra, 2 are about computers and 1 skydiving (Prof. Shenkin’s favorite hobby).

He wants to arranges the books so that the same subjects are together on the shelf. They should be arranged by probability, algebra, computers and skydiving. How many different arrangements are possible? Is Prof. Shenkin OCD?

• In R the following counts combinations and permutations: