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 Review For 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 Models 1) 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 Probability For 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 Pairs Tree Diagrams General Product Rule Permutations Combinations

Permutation An “ordered” 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 P k,n P k,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 C k,n C k,n = ( n k ) = 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).

Example A 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 Probability For 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 Rule Four 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 Probability Experiment = 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) =.30P(S A |A) =.60 P(A’)=.70P(S A’ |A’)=.10

Bayes’ Theorem A 1,A 2,….,A k a collection of k mutually exclusive and exhaustive events with P(A i ) > 0 for i = 1,…,k. For any other event B for which P(B) > 0: P(A p |B) = P (A p  B) / P(B) = P(B|A p ) P(A p )  P(B|A i ) P(A i )

Example: Bayes’ Theorem The 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

Independence Two 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 Y B: Brand X is ranked Best C: Brand X is ranked Second D: 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?

Independence Consider the following 3 events in the toss of a single die: A: Observe an odd number B: Observe an even number C: 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?