Chapter Two

Probability

Probability Definitions 

Experiment:Process that generates

observations.

Sample Space:Set of all possible

outcomes of an experiment.

Event Definitions Event: Subset of outcomes contained in the sample space. 

Simple Event: Consists of exactly one outcome.

Compound Event: Consists of more than one outcome.

Set Notation ReviewFor Two Events A and B:

Union: “A or B” = A B

Intersection: “A and B” = A B

Complement: A´Mutually Exclusive:

No outcomes in common

Probabilistic Models1) Equally Likely:

  Based on Definition  Games of Chance

2) Relative Frequency Objective Interpretation Based on Empirical Data

3) Personal Probability Subjective

Interpretation Based on Degree of

Belief

Properties of ProbabilityFor any Event A:

P(A) = 1 – P(A)

If A and B are Mutually Exclusive, P(A B) = 0 

For any two events A and B:P(A B) = P(A) + P(B) – P(A B) 

Counting Techniques 

Product Rule for Ordered PairsTree DiagramsGeneral Product Rule PermutationsCombinations

PermutationAn “ordered” arrangement ofk distinct objects taken from a set of n distinct objects.

The number of ways of ordering n distinct objects taken k at a time is Pk,n

Pk,n = n! / (n-k)!

Combination An “unordered” arrangement of k distinct objects taken from a set of n distinct objects.

The number of ways of ordering n distinct objects taken k at a time is Ck,n

Ck,n = (nk) = n! / k!(n-k)!

Example:Twenty Five tickets are sold in a lottery, with the first, second, and third prizes to be determined by a random drawing. Find the number of different ways of drawing the three winning tickets.

Example:Twenty tickets are sold in a lottery, with 5 round trips to game 1 of the World Series to be determined by a random drawing. Find the number of different ways of drawing the five winning tickets.

Example: A solar system contains 6 Earth-like planets & 4 Gas Giant-like planets. How many ways may we explore this solar system if our resources allow us to only probe 3 Gas Giants and 3 Earth-like planets?

Example:There are 50 students in ISE 261. What is the probability that at least 2 students have the same birthday? (Ignore leap years).

ExampleA dispute has risen in Watson Engineering concerning the alleged unequal distribution of 10 computers to three different engineering labs. The first lab (considered to be abominable) required 4 computers; the second lab and third lab needed 3 each. The dispute arose over an alleged ISE 261 random distribution of the computers to the labs which placed all 4 of the fastest computers to the first lab. The Dean desires to known the number of ways of assigning the 10 computers to the three labs before deciding on a course of action. What is the Dean’s next question?

Conditional ProbabilityFor any two events A and B with P(B) > 0, the conditional probability of A given that B has occurred is defined by:  P(A|B) = P(A B)/P(B)

Multiplication Rule

P(A B) = P(A|B) x P(B)

Multiplication RuleFour students have responded to a request by a blood bank. Blood types of each student are unknown. Blood type A+ is only needed. Assuming one student has this blood type; what is the probability that at least 3 students must be typed to obtain A+?

Conditional ProbabilityExperiment = One toss of a coin.If the coin is Heads; one die is thrown. Record Number.If the coin is Tails; two die are thrown. Record Sum. 

What is the Probability that the recorded number will equal 2?

Conditional Probability Problem:30% of interstate highway accidents involve alcohol use by at least one driver (Event A). If alcohol is involved there is a 60% chance that excessive speed (Event S) is also involved; otherwise, this probability is only 10%. An accident occurs involving speeding! What is the probability that alcohol is involved?

P(A) = .30 P(SA|A) = .60

P(A’)= .70 P(SA’|A’)= .10

Bayes’ Theorem

A1,A2,….,Ak a collection of k mutually exclusive and exhaustive events with P(Ai) > 0 for i = 1,…,k. For any other event B for which P(B) > 0:

P(Ap|B) = P (Ap B) / P(B) =

P(B|Ap) P(Ap) P(B|Ai) P(Ai)

Example: Bayes’ TheoremThe probabilities are equal that any of 3 urns A1, A2,& A3 will be selected. Given an urn has been selected & the drawn ball is black; what is the probability that the selected urn was A3? A1 contains: 4 W & 1 Black A2 contains: 3 W & 2 Black A3 contains: 1 W & 4 Black

IndependenceTwo events A and B are independent if:

P(A|B) = P(A)Or

P(B|A) = P(B)Or

P(A B) = P(A) P(B)and are dependent otherwise.

Independence Example:Three brands of coffee, X, Y,& Z are to be ranked according to taste by a judge. Define the following events as:

A: Brand X is preferred to YB: Brand X is ranked BestC: Brand X is ranked SecondD: Brand X is ranked Third

If the judge actually has no taste preference & thus randomly assigns ranks to the brands, is event A independent of events B, C, & D?

IndependenceConsider the following 3 events in the toss of a single die:A: Observe an odd numberB: Observe an even numberC: Observe an 1 or 2

Are A & B independent events?Are A & C independent events?

Example:A space probe to Mars has 35 electrical components in series. If the mission is to have a reliability (probability of success) of 0.90 & if all parts have the same reliability, what is the required reliability of each part?