Probability Chapter 2 A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes. The world is an uncertain place Chapter 2 A
Sections 2 -1 thru 2 -2 Let’s get started!
Random Experiment ¢ An experiment or process that can result in different outcomes each time it is repeated or observed. l l l tossing a pair of dice observing time between failures of a machine counting the number of defects found per 1000 units produced measuring the diameter of a wire produced to specifications inspecting the quality of a finished product
Basic Theme Predicting future outcomes of random processes can be very non-intuitive What random processes ¢ Therefore a method (i. e. a are you familiar with? mathematical system) is needed to quantify their likelihoods. ¢
Some Random Processes ¢ ¢ ¢ ¢ ¢ The weather – is it going to rain? Life – a birth – when? will it be a boy or a girl? Death – when am I going to die? Illness – will I catch the flu this season? Exams– will I pass the prob/stat exam? Sports – will UD win its next basketball game? Lottery and gambling – will I win the Ohio lottery if I play tomorrow? Drug Testing – will I be selected? Will I test positive? Investments – will I lose money this year on my stock holdings? Reliability – when is this car going to break down again?
Sample Space The set of all possible outcomes of a random experiment. ¢ Denoted by S. ¢ Tossing a die: S = {1, 2, 3, 4, 5, 6} l Observing time to next failure: S = {x| x 0} l Counting defects among 1000 parts produced: s = {x|x = 0, 1, … , 1000} l Inspecting quality of a finished product: S = {good, mediocre, poor} l
More about those sample spaces ¢ A sample space is discrete if it consists of a finite (or countably infinite) set of outcomes l l ¢ Manufactured part – Acceptable or Defective – finite number of outcomes – discrete Time recording device – records hours with two decimal place accuracy (countably infinite). A sample space is continuous if it contains an interval of real numbers – whether the interval is finite or not. l l l The time taken for a light bulb to burn out. The number of gallons of gasoline consumed to travel a fixed distance The package weight of a product coming off the final production line
Events ¢ a launch event into the sample space A subset of a sample space l l l Tossing a die: S = {1, 2, 3, 4, 5, 6}; define event E, getting an even number: E = {2, 4, 6} Observing time to next failure: S = {x| x 0}; define event E as the next failure occurs within 24 hours: E = {x|x 24} Counting defects among 1000 parts produced: S = {x|x = 0, 1, … , 1000}; define event E as no more then. 5% defects: E = {x| x = 0, 1, 2, 4, 5}
2 -1 Sample Spaces and Events Basic Set Operations
Need Help on Sets? If you need a refresher on set theory, the motivated student is encouraged to download and read the set theory e. Reserve material on the course Website. http: //academic. udayton. edu/charlesebeling/ENM 500/Admin/ereserve. htm
More on Events Given a sample space S, an event E is a subset of S. ¢ The outcomes in E are called the favorable outcomes. We say that E occurs in a particular experiment if the outcome of that experiment is one of the elements of E, that is, if the outcome of the experiment is favorable. ¢
Basic set operations ¢ ¢ ¢ Assume an inspection process with the following outcomes: Pass (P), Rework (R), Scrap (C) Let E 1 = {P, R} and E 2 = {R, C} Union ( ) – set of all outcomes contained in either of two events E 1 E 2 = {P, R, C} = S Intersection ( ) – set of all outcomes contained in both events E 1 E 2 = {R} Complement (E’) – set of all outcomes in the sample space not in the event E l l E 1’ = {C} S’ = { } =
The Laws of the Algebra of Sets These are very good laws. De Morgan's Laws: (A B)' = A' B‘ Idempotent Laws: (A A) = A Associative Laws: (A B) C = A (B C) Commutative Laws: (A B)= (B A) (A B) = (B A) Distributive Laws: A (B C) = (A B) (A C) Identity Laws (A ) = A (A S) = S Complement Laws (A A’) = S (A’)’ = A (A ) = (A S) = A (A A’) = S’ = , ’ = S
Other set properties ¢ Mutually exclusive – two events having no common outcomes are said to be mutually exclusive. l E 1 E 2 = says that E 1 and E 2 are mutually exclusive
Tree Diagram as Sample Space Representation A A’ (A’, B’, C’) C’ B’ C’ B B’ B C Tips of the trees represent all of the possible combinations of outcomes. C C’ C
2 -1 Sample Spaces and Events Figure 2 -5 Tree diagram for three messages.
Venn Diagrams Shading is often used to highlight the event of interest.
Union ¢ ¢ ¢ A union of two or more events is another event that contains all outcomes contained in the previous events. Union is designated by the symbol . If A and B are events then A B represents the union of A and B The union of A and B is the set of all outcomes that are either in A or B (or both), therefore A B = {x | x A or x B}. “such that” “OR” logic
Intersection ¢ ¢ ¢ The intersection of two (or more) events are those outcomes that they have in common. Intersection is designated by the symbol . So if A and B are events then the intersection is denoted by A B. The intersection of A and B is the set of all outcomes that are common to A and B, therefore A B = {x | x A and x B} “And” logic
Mutual Exclusive Events Two event are mutually exclusive (also called disjoint) if they do not have any elements in common A B =
Complement Given the event A, then the event A' is the complement of A consisting of all elements not in A; i. e. A’ = {x| x A} A A’ S Let S = the sample space (the set of all events under discussion) Then A A’ = S and A A’ = (they are mutually exclusive) Sometimes the complement of an event A is written as Ac
An Example An engineer is selected at random within a large manufacturing company. S = {all engineers of a large manufacturing company}. Define P = event, engineer selected was promoted M = event, engineer selected has a Master’s degree Find the event: 1. Engineer was promoted and has a Master’s degree. 2. Engineer was not promoted and did not have a Master’s degree. 3. Engineer was either promoted or has a Master’s degree. 4. Engineer did not get promoted but does have a Master’s degree 5. Engineer did not get promoted or has a Master’s degree. P M Pc Mc P M Pc M
Yet Another Example S = {all students taking prob/stat, org behavior, and eng. economy} A = event, a student passes the prob/stat course B = event, a student passes the org behavior course C = event, a student passes the engineering economy course Describe the event: A B C 1. A student passes all 3 courses A B C 2. A student passes at least one of the courses 3. A student passes only prob/stat A Bc Cc 4. A student fails org behavior Bc 5. A student fails at least two courses (Ac Bc C) (Ac B Cc) (A Bc Cc) (Ac Bc Cc)
To recap – Random Processes A process having an uncertain outcome called a random event that occurs by chance and cannot be predicted with certainty. The set of all possible random events is called the sample space.
We are counting on you! Counting Techniques Combinatorics is a branch of mathematics that studies collections (usually finite) of objects that satisfy specified criteria. In particular, it is concerned with "counting" the objects in those collections (enumerative combinatorics).
2 -1 Sample Spaces and Events 2 -1. 4 Counting Techniques Multiplication Rule
Permutations ¢ ¢ ¢ A PERMUTATIONS of n objects, is all of their possible arrangements or orderings. For example, the permutation of the letters a, b, and c is: l abc acb bac bca cab cba Of primary interest is knowing the number of permutations that are available from a collection of objects.
Permutations ¢ To find the number of permutations of four objects: there are 4 ways to choose the first, l 3 ways remain to choose the second, l 2 ways to choose third, l and 1 way to choose the last. l Therefore the number of permutations of 4 different things is 4· 3· 2· 1 = 24 l
Permutations of less than all A permutation of n things taken r at a time is an ordered arrangement of r objects selected from the n objects. ¢ Given the letters a, b, c then the permutations of the 3 letters taken 2 at a time are: ¢ l ab ba ac ca bc cb
How many are there? The number of permutations of n objects taken r at a time is denoted by P(n, r) or n. Pr or Pn, r.
2 -1 Sample Spaces and Events 2 -1. 4 Counting Techniques Permutations : Example 2 -10
2 -1 Sample Spaces and Events 2 -1. 4 Counting Techniques Permutations of Similar Objects
Permutations with Repetitions Example: How many distinct signals, each consisting of 6 flags hung in a line, can be formed from 4 identical red flags, and 2 blue flags? a b c a d b
2 -1 Sample Spaces and Events 2 -1. 4 Counting Techniques Permutations of Similar Objects: Example 2 -12
Combinations ¢ ¢ ¢ A combination is an unordered selection of objects. A combination of n objects taken r at a time is any selection of r of the objects where order does not count. The number of combinations of n objects taken r at a time is denoted by
The Number of Combinations ¢ Since an ordered arrangement of r objects from among n objects is a permutation where ¢ then to ignore the order of the r objects, divide by r!, the number of ways in which each set of r objects may be ordered.
Proof by Example Given 4 objects (a, b, c, d) taken 3 at a time Combinations abc abd acd bcd Permutations abc, acb, bac, bca, cab, cba abd, adb, bad, bda, dab, dba acd, adc, cad, cda, dac, dca bcd, bde, cbd, cdb, dbc, dcb
Some Combination Examples ¢ ¢ ¢ How many committees of 3 can be formed from 8 people? l C(8, 3) = 8! / 3! = (8)(7)(6) / [(1)(2)(3)] = 56 In a deck of 52 playing cards, how many different poker hands of 5 cards are there? l C(52, 5) = 2, 598, 960 In a deck of 52 playing cards, how many ways can a spade flush (5 spades) be dealt? l C(13, 5) = 1287
2 -1 Sample Spaces and Events 2 -1. 4 Counting Techniques Combinations: Example 2 -13
2 -2 Interpretations of Probability l. Measures uncertainty l. Foundation of inferential statistics l. Basis for decision models under uncertainty
What is a probability – P(E)? Let P(E) =Probability of the event E occurring, then l 0 P(E) 1 l If P(E) = 0, then event will not occur (impossible event) l If P(E) = 1, then the event will occur, i. e. a certain event So the closer P(E) is to 1, then the more likely it is that the event E will occur?
Elementary or basic events Three interpretations of probability: 1. Empirical or relative frequency n 2. A priori or equally-likely n 3. Derived from historical data Derived using counting methods Subjective n Derived from personal judgment or an individual’s degree of belief in an outcome
Relative Frequency (empirical) Let n(E) = number of times event E occurs in n trials
Example - Relative Frequency (empirical) ¢ A coin is tossed 2, 000 times and heads appear 1, 243 times. l ¢ The company’s Web site has been down 5 days out of the last month (30 days). l ¢ P(H) = 1243/2000 =. 6215 P(site down) = 5/30 =. 16667 2 out of every 20 units coming off the production line must be sent back for rework. l P(rework) = 2/20 =. 1
A priori (equally-likely outcomes) A priori (knowable independently of experience): n(A) = the number of ways in which event A can occur n(S) = total number of outcomes from the random process
Example 1 - A priori (equally-likely outcomes) ¢ A pair of fair dice are tossed. S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)} P(a seven) = 6/36 =. 1667
Example 2 - A priori (equally-likely outcomes) ¢ 5 parts are selected from a bin containing 40 parts, 10 of which are known to be defective. What is the probability of no defects among the 5 parts selected? l l How many ways can the 5 parts be selected? That is all many outcomes in the sample space? How may of these contain no defects? That is, all many favorable outcomes?
Subjective Probabilities ¢ Subjective – personal degree of belief in the likelihood of an outcome You must buy now. This stock has a 90 percent chance of doubling in value within the year. Shady Investments Inc.
More Subjective Probabilities ¢ Subjective – personal degree of belief in the likelihood of an outcome Believe me. This car is 99 percent reliable! Probability of a failure =. 01
Compound events ¢ ¢ Formed from unions, intersections, and complements of basic events Probabilities derived from l l Axioms Definitions • Sample Space • Events • Conditional Probability l Theorems • Addition law • Multiplication law • Bayes theorem ¢ Collectively constitute the mathematics of probability
2 -2 Interpretations of Probability 2 -2. 2 Axioms of Probability
Mutually Exclusive Events S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1) (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)} ¢ ¢ ¢ A pair of dice are tossed. Find Pr{a seven is tossed or doubles are tossed} Let A = event, a seven is tossed Let B = event, doubles are tossed P(A) = 6/36 and P(B) = 6/36 P(A B) = P(A) + P(B) = 12/36 =. 3333
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