Probability and statisticsMath 381

Instructor: Maya Chatila

Grade Distribution: Midterm:40%, Final: 40%, h.w +c.w+attendance: 20%.Class Rules: -Attendance is mandatory, no one can attend in other section under any circumstances.-Any h.w submitted without the h.w report will be cancelled.-No h.w will be accepted after the deadline under any excuse.-Midterm: Friday 3/7/2015. No make up exam will be given and automatically a zero will be given if case of absence .

Book: Introductory statistics, Neil A.Weiss, 9th editionhttp://202.114.108.237/Download/4ad56511-0538-47b1-9c39-0aff5c3f6403.pdf

P A R T IIICHAPTER 4

Probability Concepts

Terminology

• Experiment: is any process that can be repeated in which the results are different or cannot be predicted with certainty.

• Sample space: is the set of all outcomes.Example: one coin toss

S = {H,T} Example: three coin tosses

S = {HHH, HTH, HHT, TTT, HTT, THT, TTH, THH}Example: roll a six-sided dice

S = {1, 2, 3, 4, 5, 6}

Sample Space might be discrete or continuous discrete

– finite number of outcomes

continuous– outcomes vary along continuous scale

Terminology

• Event is an outcome or a set of outcomes of a random process.

Example: Tossing a coin three timesEvent A = getting exactly two heads = {HTH, HHT, THH}

Example: Picking real number X between 1 and 20Event A = chosen number is at most 8 = {X ≤ 8}

Example: Tossing a fair diceEvent A = result is an even number = {2, 4, 6}

• For convenience, we use letters such as A, B, C, D, . . . to represent events. In the card-selection experiment, we might let

A = event the card selected is the king of hearts,B = event the card selected is a king,C = event the card selected is a heart, andD = event the card selected is a face card.

Probability for Equally Likely Outcomes Suppose an experiment has N possible outcomes, all equally likely. An event that can occur in f ways has probability f/N of occurring:Probability of an event = f : Number of ways event can occurN: Total number of possible outcomes

Ex1: One Die is Tossed P(even number) = |2,4,6| / |1,2,3,4,5,6|

Ex2: When two balanced dice are rolled, 36 equally likely outcomes are possible. Find the probability thata. the sum of the dice is 11.b. both dice come up the same number.c. the sum of the dice is 1.d. the sum of the dice is 12 or less.

At Least, At Most, and Between

For any numbers x and y: The phrase “at least x” means “greater than or equal to x,” The phrase “at most x” means “less than or equal to x,” The phrase “between x and y” means “greater than but less than y.”

Properties of Probabilities

• The probability of any event E, P(E), must be between 0 and 1 inclusive. That is,

0 < P(E) < 1.

• If an event is impossible, the probability of the event is 0.

• If an event is a certain, the probability of the event is 1.

• If S = {e1, e2, …, en}, then

P(e1) + P(e2) + … + P(en) = 1.

The total probability of all possible event always sums to 1.

Ex3: Suppose a “fun size” bag of M&Ms contains 9 brown candies, 6 yellow candies, 7 red candies, 4 orange candies, 2 blue candies, and 2 green candies. Suppose that a candy is randomly selected.

• (a) What is the probability that it is brown?

• (b) What is the probability that it is blue?

Ex 4: Housing Units. The U.S. Census Bureau publishes data on housing units in American Housing Survey for the United States. The following table provides a frequency distribution for the number of rooms in U.S. housing units. The frequencies arein thousands.Rooms No. of units1 6372 1,3993 10,9414 22,7745 28,6196 25,3257 15,2848+ 19,399A U.S. housing unit is selected at random. Find the probability that the housing unit obtained hasa. four rooms. b. more than four rooms.c. one or two rooms. d. fewer than one room.e. one or more rooms.

Combinations of Events (Relationships Among Events)

• The complement Ac of an event A is the event that A does not occur

• The union of two events A and B is the event that either A or B or both occurs

• The intersection of two events A and B is the event that both A and B occur

Event A Complement of A Union of A and B Intersection of A and B

Ex5: A frequency distribution for the ages of 40 students in a statistics class is presented below. One student isselected at random. LetA = event the student selected is under 21,B = event the student selected is over 30,C = event the student selected is in his or her 20s, andD = event the student selected is over 18.Determine the following events.a. (not D) b. (A & D) c. (A or D) d. (B or C)

• Two events are called disjoint or mutually exclusive if they can not happen at the same time • Events A and B are disjoint means that the intersection of

A and B is empty. • Example: coin is tossed twice • S = {HH,TH,HT,TT}• Events A={HH} and B={TT} are disjoint • Events A={HH,HT} and B = {HH} are not disjoint

Exercises

1. Consider the sample space:S = {copper, sodium, nitrogen, potassium, uranium, oxygen , zinc}And the events:A= { copper, sodium, zinc}B= {sodium, nitrogen, potassium}C= {oxygen}List the elements of the following events:, C.

Exercises

2. Consider the sample space: S = {x/0<x<12}And the events: M= { x/1<x<9} and N= {x/0<x<5}Find: .

Properties

The special addition rule states for mutually exclusive events, the probability that one or another of the events occurs equals the sum of the individual probabilities. that is, P(A or B) = P(A) + P(B).The General Addition Rule: For events that are not mutually exclusive, we must use the general addition rule.

P(A or B) = P(A) + P(B) − P(A & B).The Complementation Rule

P(Ac) = 1 - P(A)

Note:

P(A)= P(A+ P(AB)

Ex: Playing Cards Consider again the experiment of selecting one card at randomfrom a deck of 52 playing cards. Find the probability that the card selected is eithera spade or a face carda. without using the general addition rule.b. by using the general addition rule.

Ex: Records for one year show that 76.2% of the people arrested were male, 15.3% wereunder 18 years of age, and 10.8% were males under 18 years of age. If a person arrested that year is selected at random, what is the probability that that person is either male or under 18?

Homework

p.151 # 4.19 , 4.22, 4.38, 4.48, 4.50p.168 # 4.79 , 4.80, 4.82

Conditional Probability

The probability that event B occurs given that event A occurs is called a conditionalprobability. It is denoted P (B |A), which is read “the probability of B given A.” We call A the given event.

Ex: Rolling a Die When a balanced die is rolled once, six equally likely outcomes arePossible. LetF = event a 5 is rolled, andO = event the die comes up odd.Determine the following probabilities:a. P(F), the probability that a 5 is rolled.b. P(F | O), the conditional probability that a 5 is rolled, given that the die comes up odd.c. P(O | (not F))

The Conditional Probability Rule

In the previous example, we computed conditional probabilities directly, meaning that we first obtained the new sample space determined by the given event and then, using the new sample space, we calculated probabilities in the usual manner.Sometimes we cannot determine conditional probabilities directly. We obtain a formula fordoing so.

Conditional Probability

Ex: imperfect diagnostic test for a disease

• What is probability that a person has the disease? Answer: 40/100 = 0.4

• What is the probability that a person has the disease given that they tested positive?More Complicated !

Disease + Disease - Total

Test + 30 10 40

Test - 10 50 60

Total 40 60 100

Definition: Conditional Probability

• Let E and F be two dependent events in sample space. The conditional probability that event E occurs given that event F has occurred is:

• Ex: probability of disease given test positive

P(disease +| test +) = P(disease + and test +) / P(test +) = (30/100)/(40/100) =.75

)(

)()|(

FP

FEPFEP

Ex: Coin Tossing. A balanced dime is tossed twice. The four possible equally likely outcomes are HH, HT, TH, TT. Let A = event the first toss is heads, B = event the second toss is heads, and C = event at least one toss is heads.Determine the following probabilities:a. P(B) b. P(B | A) c. P(B |C)d. P(C) e. P(C | A) f. P(C | (not B))

Ex : A math teacher gave her class two tests. 25% of the class passed both tests and 42% of the class passed the first test. What percent of those who passed the first test also passed the second test?

The General Multiplication Rule

The General Multiplication Rule :If A and B are any two events, then P(A & B) = P(A) * P(B | A).

Ex: For the 110th Congress, 18.7% of the members are senators and 49% of the senatorsare Democrats. What is the probability that a randomly selected member of the 110th Congress is a Democratic senator?

Ex: The table below shows the gender frequency of a class of 40 students Male 17 Female 23Two students are selected at random from the class. The sampling is without replacement.Find the probability that the first student selected is female and the second is male.Let F1 = event the first student obtained is female, and M2 = event the second student obtained is male.We want to determine P(F1& M2).P(F1 & M2) = P(F1) · P(M2 | F1) = . = 0.251.

Independent Events

Independent Events: Event B is said to be independent of event A if P(B | A) = P(B).The Special Multiplication Rule: If A and B are independent events, then P(A & B) = P(A) · P(B)and conversely, if P(A & B) = P(A) · P(B), then A and B are independent events.

We can decide whether event A and event B are independent by using either of two methods. we can determine whether P(B | A) = P(B) or P(A & B) = P(A) · P(B).

Note: The terms mutually exclusive and independent refer to different concepts. Mutually exclusive events are those that cannot occur simultaneously; independent events are those for which the occurrence of some does not affect the probabilities of the others occurring.In fact, two events cannot be both mutually exclusive and independent.

Exercises

1. Let A and B be events such that p(A)=1/4, p(A1/3. Find p(B) if:

a) A and B are independent.b) A and B are mutually exclusive.c) A is a subset of B.

2. Calculate P(Af P(A)=0.4, P(B)=0.45 and P(A=0.1.

3. What is the probability that neither A nor B occur if P(=0.6 and P(=0.5.

Exercises

4.Let A and B be events such that P(A)=1/2, P(B)=1/3 and P(A=7/12. Find P(A/B), P(A/), P(/B) and P(/).5. In a class of 100 students, 54 studied math, 69 studied English and 35 studied both math and English. If one of these students is selected at random, find the probability that the student:

a) Took either math or English.b) Took English but not math.c) Took both subjectsd) Took neither math nor English.

Exercises

6. On 30 students in a class, 25 students speak English and 20 students speak French. Each students speak at least one of these languages. We meet a student from this class. What is the probability that this student speaks:a) Both languagesb) French onlyc) One language only

Bayes’s Rule

The Rule of Total Probability: Suppose that events A1, A2, . . . , Ak are mutually exclusive and exhaustive; that is, exactly one of the events must occur. Then for any event B,P(B) =

Bayes’s Rule : Suppose that events A1, A2, . . . , Ak are mutually exclusive and exhaustive. Then for any event B,• P(Ai | B) =

Ex : 7.0% of the population has lung disease. Of those having lung disease, 90.0% are smokers; of those not having lung disease, 25.3% are smokers. Determine the probability that a randomly selected person is:a) A smokerb) A smoker has lung disease.

Exercises

Ex: Principals rent cars from three agencies: 60% from A1, 30% from A2, and 10% from A3. If 9% of the cars from A1 are defective, 20% from A2 are defective and 6% of the cars from A3 are defective.a) what is the probability that a rental car will

be defective?b) If the rental car is defective, what is the

probability that it came from A2?