STA 4321 Exam #1

Distributive Laws: A ∩ (B ∪ C) = ? A ∪ (B ∩ C) = ?

(A ∩ B) ∪ (A ∩ C) (A ∪ B) ∩ (A ∪ C)

*Counting Rules:* What are the two most important aspects when deciding which rule to use?

1. whether were replacing or not replacing 2. whether order matters

*Computer Terminals Example:* A manufacturer has five seemingly identical computer terminals available for shipping. Unknown to her, tow of the five are defective. A particular order calls for two of the terminals and is filled by randomly selecting two of the five that are available. Find the probability of event A

Because A = E8 U E9 U e10, axiom 3 implies that P(A) = P(E8) + P(E9) + P(E10) = 3/10

Events are denoted by ?

Capital Letters A, B, C, etc.

Two events A and B are said to be independent if:

P(A ∩ B) = P(A)P(B) P(A | B) = P(A) P(A | B) = P(B) if P(A) = 4/52 P(B) = 13/52 and P(A ∩ B)= 1/52 then A and B are said to be independent A and are independent if the probability of he occurrence of event A is is unaffected by the occurrence or nonoccurence of event B

1 - P(Ā) = ?

P(A) P(A) = 1 - P(Ā) It is sometimes easier to calculate P(Ā) than to calculate P(A)

What does each simple event correspond to?

corresponds to one and only one sample point

What is the set of a possible sets?

denoted by a curvy A example: Choose from 4 people at random S = {1, 2, 3, 4} Curvy A = { null set, {1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3} .......{1,2,3,4} etc.}

What is are events?

the outcomes of an experiment if the experiment consists of counting the number of bacteria in a portion of food, some events could be: A: exactly 110 bacteria are present B: more than 200 bacteria are present C: the number of bacteria present is between 100 and 300

the probability of any event is the sum of ?

the probabilities of the sample points in the given event example: S = ({1, 2, 3, 4}) A = event that person 1 or person 2 is chosen P(A) = P({1, 2} = P({1}) + P({2}) = 1/4 + 1/4 = 1/2 the procedure of providing a probability assignment P, guarntess that the probability assignment satsfies the 3 axioms, for discrete sample spaces

What is the probability mass function

the probability y that Y takes on the value y, P(Y=y) The probability mass function is a random variable x (discrete), is given by px(x) = triangle P(x = x) for every x ∈ squiggly X

For any discrte probabiltiy distribtuon, the following must be true:

1. 0 <= p(y) <= 1 for all y 2. sum notation y p(y)=1, where the summation is over al lvalues of y wit hnonzero probability

Summary of the steps used in the Event-Composition Method:

1. Define the experiment 2. Visualize the nature of the sample points. Identity a few to clarify your thinking. 3. Write an equation expressing the event of interest as a composition of two or more events, using unions, intersections, and/or complements. Make sure that event A and the event implied by the composition represent the same set of simple events. 4. Apply the additive and multiplicative laws of probability to the compositions obtained in step 3 to find P(A) Step 3 is the most difficult because we can form many compositions that will be equivalent to event A. The trick is to form a composition in which all the probabilities appearing in Step 4 are known.

Sample-point method: The following steps are used to find the probability of an event using this method:

1. Define the experiment and clearly determine how to describe one simple event 2. List the simple events associated with the experiment and test each to make certain it cannot be decomposed. This defines the sample space S. 3. Assign reasonable probabilities to the sample points in S, making certain that P(Ei)>= 0 and sum function P(Ei) = 1 4. Define the event of interest, A, as a specific collection of sample points. Test all sample points in S to identify those in A 5. Find P(A) by summing the probabilities of the sample points in A.

What are the axioms of probability?

1. P(A) ≥ 0 for every event A 2. P(S) = 1 3. If A1, A2, A3 ...... is a sequence of mutually exclusive events (i.e., Ai ∩ Aj = ∅ for every i ≠ j), then: P( infinity U i =1 Ai) = infinity sum notation i = 1 P(Ai) basically if A ∩ B = ∅, then P(A U B) = P(A) + P(B)

The number of unordered subsets of size r chosen without replacement from n available objects is calculated by ?

= n! / [r!(n-r)!] The selection of r objects from a total of n is equivalent to partitioning the n objects into k =2 groups, the r selected, and the (n-r) remaining. This is a special case of the general partitioning problem. In the present case, k = 2, n1 = r, and n2 = (n-r) example: Let A denote the event that exactly one of the two best applicants appears in a selecting out of the five. Find the number of simple events in A since we are selecting one applicant that is the best and the other applicant will be the worst then we can divide the applicants up into two sections section one will be the two best applicants and section two will be the three worst applicants within section 1 there are two groups. The "group" of best applicant that is selected and the "group" of best applicant that is not selected n = 2 since there are two best applicants n1 = r and since we are selecting one specific object r = 1 n2 = (n-r) and will equal 1 basically n1 is the "group" of selected best applicant and n2 is the "group" of unselected best applicant section 1 = 2! / (1!@1!) within section 2 there are two groups. The "group" of worst applicant that is selected and the "group" of worst applicant that is not selected n = 3 since there are three worst applicants n1 = r and since we are selecting one specific object r = 1 n2 = (n-r) and will equal 2 basically n1 is the "group" of selected worst applicant and n2 is the "group" of unselected worst applicant section 1 = 3! / (1!@2!) Then apply the mn rule and multiply section 1 and section 2 together [2! / (1!@1!)] @ [3! / (1!@2!)] = 6

A and B are called mutually excluive if ?

A ∩ B = ∅ the empty set

Examples of events if a single toss of a balanced die are:

A: observe an odd number B: observe a number less than 5 C: observe a 2 or 3

What is the law of total probability?

Assume that {B1, B2, .... Bk} is a partition of S such that P(Bi) > 0, fr i = 1, 2, .... , k Then for any event A: P(A) = k sum notation i =1 P(Bi)P(A|Bi) =P(A|B1)P(B1) + P(A|B2)P(B2)+.....+P(A|Bk)P(Bk) example: A company buys microchips from three suplliers. Supplier 1 microchips have 10% chance of being defective, Supplier 2 microchips have 5% chance and Supplier 3 microchips have 2% chance of being defective. Suppose 20%, 35%, and 45% of the current supply came from Suppliers 1, 2, and 3 respectively. If a microchips is selected randomly from this supply what si the probability that it is defective? B1 = event that microchip came from supplier 1 B2 = event that microchip came from supplier 2 B3 = event that microchip came from supplier 3 A - event that microchip is defective P(A|B1) = 0.1, P(A|B2) = 0.05, P(A|B3) = 0.02 P(B1) = 0.02, P(B2)= 0.35, P(B3) = 0.02 Hence, by the theorem of total probability P(A) = [email protected] + [email protected] + [email protected] = 0.046

*Counting Rules:* number of ways of choosing r specific objects from n total objects *Order does not matters without replacement*

Combinations: C^n r = n! / r!(n-r)!

What is used to denote simple events or the corresponding sample point?

E with a subscript eg- E1 or E7 etc

Let Y be a discrete rando mvariabe with the probability fucntion p(y). Then the expected value of Y, E(Y) is defined to be

E(Y) = sum notation y p(y)

Let Y be a discree random variable wit hthe probability fucntion p(y) and g(Y) be a rea lvlaued fucntion of Y the nthe expected value of g(Y) is given by

E[g(Y)] = sum notation all y g(y)p(y)

*Computer Terminals Example:* A manufacturer has five seemingly identical computer terminals available for shipping. Unknown to her, tow of the five are defective. A particular order calls for two of the terminals and is filled by randomly selecting two of the five that are available. Let A denote the event that the order is filled with two non defective terminals. List the sample points in A

Event A = {E8, E9, E10}

What is a partition of S?

For some positive integer k, let the sets B1, B2, ..... Bk be such that 1. S = B1 U B2 U .... U Bk 2. Bi ∩ Bj = ∅, for i ≠ j Then the collection of sets {B1, B2, ...., Bk} is said to be a partition of S

The distribution function F for a random variable x is defined by:

Fx(b) = P(x <= b) for all b in real numbers if x is a discrete random variable, Fx(b) = P(x <= b) = sum notation x<= b Px(x) it is important to study a random variable by looking at its cumulative properties

What is the Bayes Rule?

If the events B1, B2, ........, Bk are mutually exclusive and exhaustive, then for any event A, P(Bi|A) = [P(A|Bi)P(Bi)] / k sum notation j = 1 P(A|Bj)P(Bj) In the example about defective microchips, if a randomly selected microchip is defective, what is the probability that it came from supplier 2? Bayes Rule: P(B2|A) = [P(A|B2)P(B2)] / 3 sum notation i =1 P(A|Bi)P(Bi) =0.376

What does it mean to be mutually exclusive?

In logic and probability theory, two events are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails, but not both.

*Computer Terminals Example:* A manufacturer has five seemingly identical computer terminals available for shipping. Unknown to her, tow of the five are defective. A particular order calls for two of the terminals and is filled by randomly selecting two of the five that are available. List the sample space for this experiment

Let the two defective terminals be labeled D1 and D2 and let the three good terminals be labeled G1, G2, and G3. Any single simple events (sample point) will consist of a list of the two terminals selected for shipment. The simple events may be denoted: E1 = {D1, D2} E2 = {D1, G1} E3 = {D1, G2} E4 = {D1, G3} E5 = {D2, G1} E6 = {D2, G2} E7 = {D2, G3} E8 = {G1, G2} E9 = {G1, G3} E10 = {G2, G3} Thus there are ten simple events(sample points) in S, and S={E1, E2, ...... E10}

How can you calculate the number of ways you can partition n objects into k groups ?

N = n! / (n1!@[email protected]!) n = the amount of objects you have n1 = the amount of objects you can have in group 1 n4 = the amount of objects you can have in group 4 n3 n6 n100 etc. k = the number of groups you have N = is basically the number of simple events (number of distinct arrangements) example: if you have 4 people that need to be placed in two groups how many simple events can you create? a person b person c person d person E1 = ab | cd E2 = ac | bd E3 = ad | bc E4 = bc | ad E5 = bd | ac E6 = cd | ab 6 simple events of N = 6 N = n! / (n1! @ n2!) N = 4! / (2! @ 2!) N = 24 / 4 N = 6

Conditional Probability: The probability of the event A given that the event B has occurred is defined as:

P(A | B) = P(A ∩ B) / P(B) P(A | B) is read as the probability of A given B example: Suppose that a balanced die is tossed once. Find the probability of a 1, given that an odd number was obtained A ⊆ B so A ∩ B = A and P(A ∩ B) = P(A) = 1/6 P(B) = 1/2 P(A | B) = P(A ∩ B) / P(B) (1/6) / (1/2) = 1/3

Counting Rule #1: Suppose we are performing an experiment where all outcomes are equally likely, hence we assign the same probability to every sample point (say there are N sample points) Suppose the event of interest say A consist of nA sample points. then the P(A) is:

P(A) = nA / N P(A) = # sample points in A / # total sample points

*Counting Rules:* number of ways of choosing r specific objects from n total objects *Order matters without replacement*

Permutation: P^n r = n! / (n-r)!

How are sets denoted?

Sets are denoted by capital letters e.g.- A, B, C....

Demonstrate the Proof that P(∅) = 0

Shown p.7

What is the Event-Composition Method of Calculating the Probability of an Event?

Since sets (events) can often be expressed as unions, intersections, or complements of other sets, the event-composition method for calculating the probability of an event, A, expresses A as a composition involving unions and/or intersections of other events. The laws of probability are then applied to find P(A) Example: Of the voters in a city, 40% are Republicans and 60% are Democrats. Among the Republicans 70% are in favor of a bond issue, whereas 80% of the Democrats favor the issue. If a voter is selected at random in the city, what is the probability that he or she will favor the bond issue Let F denote the event "favor the bond issue" R the event "A Republican is selected" and D the event "a Democrat is selected" Then P(R) = .4, P(D) = .6, P(F|R) = .7, and P(F|D)=.8 P(F) = P[(F∩R) U (F∩D)] = P(F∩R) + P(F∩D) P(F∩R) = P(F|R)P(R) = (.7)(.4) = .28 P(F∩D) = P(F|D)P(D) = (.8)(.6) = .48 P(F) = .28 + .48 = .76

Let Y be a discrete random variable with probability function p(y) and mean E(Y) = mu, then:

V(Y) = σ^2 = E[(Y-mu)^2] = E(Y^2) - mu^2 this theorem greatly reduces the labor in finding the variance of a discrete random variable

*Counting Rules:* What does "with replacement" mean? What does "without replacment" mean? What does order matter? What does order doesnt matter?

With replacement means the same item can be chosen more than once. Without replacement means the same item cannot be selected more than once. whether the order of objects selected matters example of when order matters is a PIN code The PIN Code 1234 is different from the PIN Code 4321

What does Y and y denote?

Y denotes a random variable y denotes a particular value that a random variable may assume eg Y denote any one of the six possible values that could be observed on the upper face when a die is tossed and y denotes the number actually observed

What is a point?

a distinct object within a set {1, 2, 3} 1 would be a point

What is a probability assignment *P*?

a numerically valued function that assigns a value P(A) to every event A so that the axioms of probability are satisfied

What is a discrete random variable?

a random variable Y is said to be discrete if it can assume only a finite or countably infinite number of distinct values eg the number of bacteria per unit area in the study of drug control on bacterial growth

What is a random variable?

a random variable is a variable whose values depend on outcomes of a random phenomenon example: Let H and T represent head and tail, respectively; and let an ordered pair of symbols identify the outcome for the first and second coins. (Thus HT implies a head on the first coin and a tail on the second.) Then the four sample points in S are E1: HH, E2: HT, E3: TH, and E4: TT. The values of Y assigned to the sample points depend on the number of heads associated with each point. For E1: HH, two heads were observed and E1 is assigned the value Y = 2. Similarly, we assign the values Y = 1 to E2 and E3 and Y = 0 to E4. Summarizing, the random variable Y can take three values, Y = 0, 1, and 2, which are events define d by specific collections of sample points: {Y=0} = {E4} {Y=1} = {E2, E3} {Y=2} = {E1} Notationally, X: S ----> all real numbers Example: Experiment: Sample n people from a population of N people Random Variable: the average height the set of possible values that a random variable X takes is denoted by squiggly X

What is a set?

a set is a collection of distinct objects, and the distinct objects within the set are referred to as *elements* or *points* {1, 2, 3} is a set 1 would be a point

What is a compound event?

an event that can be decomposed into other events example: if you are flipping a coin and you have A: observe an odd number E1: observe a 1 E3: observe a 3 E5: observe a 5 and you observe event A then at the same time you will have observed E1, E3, and E5 E1, E3, and E5 cannot be broken down into smaller events and are therefore referred to as simple events

What is a simple event?

an event that cannot be decomposed or broken down into smaller events if you are flipping a coin and you have A: observe an odd number E1: observe a 1 E3: observe a 3 E5: observe a 5 and you observe event A then at the same time you will have observed E1, E3, and E5 E1, E3, and E5 cannot be broken down into smaller events and are therefore referred to as simple events

The process by which an observation is made is ?

an experiment examples: coin and die tossing, measuring the IQ score of an individual, determining the number of bacteria per cubic centimeter in a portion of processed food

What is a permutation?

an ordered arrangement of *r* distinct objects the number of ways of ordering n distinct objects taken r at a time will be designed by the symbol P^n subscript r

*Computer Terminals Example:* A manufacturer has five seemingly identical computer terminals available for shipping. Unknown to her, tow of the five are defective. A particular order calls for two of the terminals and is filled by randomly selecting two of the five that are available. assign probabilities to the simple events

because the terminals are selected at random, any pair of terminals is as likely to be selected as any other pair. Thus, P(Ei) = 1/10, for i =1, 2,......, 10 is a reasonable assignment of probabilities

What does P^n subscript r equal?

n! / (n-r)! n = objects r= specific objects P^n subscript r = basically how many simple events can you make Example: you have 30 employees and you will draw three names of employees from a hat to be given a prize. 1st place is $100 2nd place is $50 and 3rd place is$20 so order matters. How many simple events are there? mn rule method: there are 3 stages (each time you draw a name) but each stage has different ways (number of names) when you draw a name you don't replace it so then the 30 names become 29 names for stage 2 and so on therefore stage 1 has 30 options, stage 2 has 29 options, and stage 3 has 28 options 30 @ 29 @ 28 = 24,360 or you could do it by using the permutation formula there are 30 objects (names) so n = 30 and you are selecting three specific objects (three specific names) from the thirty so 30! / (30-3)! = 24,360 if you expand the factorial you can see that all the numbers in the numerator will be canceled out by the numbers in denominator except for 30, 29, and 28 which shows the relationship between the two methods

*Counting Rules:* number of ways of choosing r specific objects from n total objects *Order matters with replacement*

n^r

Sets are comprised of ?

sample points

Demonstrate the Proof that P(A) ≤ P(B) if A ⊆ B

shown p.7

Demonstrate the Proof that P(A) + P(B) if A∩B = ∅

shown p.7

What does Ā stand for?

stands for complement which is the collection of all points in S that is not in set A

What does ∩ stand for?

stands for intersection A ∩ B means the collection of points which are in A and B

What does ∅ stand for?

stands for null or empty set a set with no points

What does ⊆ stand for?

stands for subset A ⊆ B means that a is a subset of B, i.e. all points in A are also in B

What does ∪ stand for?

stands for union A ∪ B means the collection of points which are in A or B or both

What does S stand for?

stands for universal set (sample space) which is the collection of all points of interest in the current situation

If Y is a rando mvariable wit hmean E(Y) = mu, the variance of a rando mvariable Y is ?

the expected value of (Y-mu)^2 V(Y) = E[(Y-mu)^2] The standard deviation of Y is the positive square root of V(Y) population variance denoted by σ^2 population standard deviation denoted by σ

What is the Fundamental Principle of Counting (mn rule)?

the fundamenta lrpinciple of counting is basically the mn rule if an experiment is performed in two stages with m ways to accomplish the first stage and n ways to accomplish the second stage, then there are mn ways to accomplish the experiment this can be extended 3 ways, 4 ways, etc. example: we ask 20 people what their birthday is each simple event would comprise the 20 dates that the people would provide so each simple event is a 20-tuple basically how many 20-tuples could we provide? well if there are 365 days in a year then the first person has 365 options to choose from so m = 365. the second person also has 365 options/ways to choose from so n = 365 and so on until we reach the 20th person. so there are 20 "stages" and 365 "ways" and since its just 365(1st person) @ 365(2nd person) @ 365(third person)....... 365(20th person) the answer would be: 365^20

What is the inclusion-exclusion principle (additive law of probability)? What is the formula?

the inclusion-exclusion principle provides an identity for computing the probability of union of a set of events in terms of intersections of various orders of these events the probability of the union of two events A and B is: Two events: P(AUB) = P(A) + P(B) - P(A∩B) If A and B are mutually exclusive events, P(A∩B) = 0 and P(AUB) = P(A) + P(B) Three events: P(A) + P(B) + P(C) - P(A∩B) - P(B∩C) - P(A∩C) + P(A∩B∩C) k events: If A1, A2, ....., Ak are k events, P(A1UA2U........UAk) = k sum notation i =1 P(Ai) - sum notation all unordered pairs (i1, i2) P(Ai1 ∩ Ai2) + sum notation all unordered triplets (i1, i2, i3) P(Ai1 ∩ Ai2 ∩ Ai3) - ........ + (-1)^(k-1) P(Ai1 ∩ Ai2 ∩ ...... ∩Aik) gives the probability of the union of two events

What are combinations?

the number of combinations of n objects taken r at a time is the number of subsets, each of size r, that can be formed from the n objects. This number will be denoted by C^ n subscript r or (n r) n is on top of r

what does n subscript a stand for?

the number of simple events in event A

What is an event?

the outcome(s) of an experiment events are demoted by capital letters any collection of sample points. In other words, any subset of the sample space is called an event example: toss a coin 3 times S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT} A = event that there is at least on head A = {HHH, HHT, HTH, THH, TTH, THT, HTT}

What is the probability distribution?

the probability distribution for a discrete variable Y can be represented by a formula, a table, or a graph that provides p(y) = P(Y=y) for all y example: A supervisor in a manufacturing plant has three men and three women working for him. He wants to choose two workers for a special job. Not wishing to show any biases in his selection, he decides to select two workers at random. Let Y denote the number of women in his selection. Find the probability distribution for Y The supervisor can select two workers from six in (6 2) = 15 ways. Hence, S contains 15 sample points, which we assume to be equally likely because random sampling was employed. Thus, P(Ei) = 1/15, for i = 1, 2,.... 15. The values for Y that have nonzero probability are 0, 1, 2. The number of ways of selecting Y = 0 women is (3 0 )(3 2) because the supervisor must select zero workers from the three women and two from the three men. Thus, there are (3 0)(3 2) = 3 simple events(sample points) in the event Y = 0 and therefore the probability distribution for Y is: p(0) = P(Y=0) = 1/5 p(1) = P(Y=1) = 3/5 p(2) = P(Y=2) = 1/5

What is the Multiplicative Law of Probability?

the probability of the intersection of two events A and B is P(A∩B) = P(A)P(B|A) = P(B)P(A|B) If A and B are independent, then P(A∩B) = P(A)P(B) The multiplicative law follows directly from the definition of conditional probability The multiplicative law can be extended to find the probability of the intersection of any number of events The probability of the intersection of any number of k events can be obtained by: P(A1∩A2∩A3∩.....∩Ak) = P(A1)P(A2|A1)P(A3|A1∩A2)........P(Ak|A1∩A2∩....∩Ak-1) gives the probability of intersection of events

What is the sample space?

the set consisting of all possible sample points/simple events denoted by an S

What is the sample space S of a random experiment is ?

the set of all possible outcomes of the experiment listed in a mutually exclusive and exhaustive way the set consisting of all possible sample points. a sample space will be denoted by S discrete or countable sample space example: toss a coin 4 times S = {HHHH, HHHT, HHTT, .......} = 2^4 = 16 possible outcomes continuous or uncountable sample space example: percentage of a population affected by an epidemic S = [0, 100] all real numbers from 0 to 100

DeMorgan's Laws: (A ∩ B) bar = ? (A ∪ B) bar = ?

Ā ∪ B̄ Ā ∩ B̄