Chapter 5 Stats

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An event may consist of...

one outcome or more than one outcome

What's considered unusual?

outside 2 standard deviations OR is 5% & below

Example 6: In a national survey conducted by the Centers of Disease Control in order to determine college students' health risk behaviors, college students were asked "How often do you wear a seatbelt with a car?" The frequencies were recorded below: RESPONSE Frequency Never 118 Rarely 249 Sometimes 345 Most of the Time 716 Always 3,093 (b) Is it unusual for a college student to never wear a seatbelt in a car? Why?

since p(never) = 3% (close enough to 0), it's unusual for a student to never wear a seatbelt.

Sample space

the collection of all possible outcomes

Complement is denoted by

A^c (but i'm lazy so i just do Ac)

Example 5: Suppose the experiment is rolling a pair of fair dice (a) Write the sample space of the experiment --> how many outcomes? --> smallest sum? --> largest sum?

# of outcomes: 6 • 6 = 36 - smallest sum = 2 - largest sum = 12

Example 4: Suppose we are flipping a coin and rolling a die. What is the probability that we flip a tail and roll a 3?

(1/2) • (1/6) = 1/12

Example 3: Suppose we are rolling a die two times. What is the probability that we roll a 6 both times?

(1/6) • (1/6) = 1/36

Example 5: Suppose the experiment is rolling a pair of fair dice (c) Find the probability that the sum is greater than 8

(3, 6) | (4, 5) | (4, 6) | (5, 4) | (5, 5) | (5, 6) | (6, 3) | (6, 4) | (6, 5) | (6, 6) sum = 9, 10, 11, 12 P (sum greater than 8) = 10/36 = 5/18

Example 5: Suppose the experiment is rolling a pair of fair dice (b) Find the probability that the sum is 10

(5, 5) (4, 6) (6, 4) P(sum of 10) = 3/36 = 1/12

classical method in babie

not going to collect data. just @ home and compute and predict anyway

Example 5: You receive a shipment of 12 TVs. Three are defective. a) If you randomly choose two TVs, what is the probability that both of them work?

0.545

Example 1: Suppose our probability experiment consists of rolling a single fair die. b) Define the event E = rolling an odd number

* all the possible outcomes of THAT event E = {1, 3, 5}

Example 3: The following probability model shows the distribution for the number of rooms in US housing units. Rooms Probability 1 0.005 2 0.011 3 0.088 4 0.183 5 0.230 6 0.204 7 0.123 8 or more 0.156 (a) Verify that this is a probability model

*check: • negative probabilities --> none ✓ • probabilities are less than 1 --> ✓ • total prob = 1 (needs to be exact) --> ✓

Rules of Probabilities 1. Probabilities are between _______. That is, for any event E, ________

- 0 and 1 - 0 ≤ P(E) ≤ 1

Rules of Probabilities 2. If an event is impossible, then the probability of that event is ___

Rules of Probabilities 3. If an event will definitely happen, then the probability of that event is

Rules of Probabilities 5. The sum of the probabilities of all outcomes must equal __.

P(at least one is) =

1 - P(all are not)

P(at least one is not) =

1 - P(all are)

An experiment consists of rolling a die and then tossing a coin. Find the sample space for this experiment.

1 < H, T 2 < H, T 3 < H, T 4 < H, T 5 < H, T 6 < H, T rolling a die (6) • tossing a coin (2) = 12 outcomes S = {1H, 1T, 2H, 2T, 3H, 3T, 4H, 4T, 5H, 5T, 6H, 6T}

Example 7: Suppose that A and B are two events and P(A) = 0.5, P(B) = 0.7, and P(A | B) = 0.5 Are A and B independent?

P(A | B) = P(A) 0.5 = 0.5 --> A and B are independent

how many cards in a deck of cards, , how many red?, how many black? how many suites, how many cards make up the suite??

52 cards 26 black & 26 red 4 suites: 13 diamond, 13 hearts, 13 clubs, 13 spades

(a) Suppose we grab the 1st ball out of the bag, which is a 3. Then, we leave that ball out of the bag and grab the second ball. (b) Suppose we grab the 1st ball out of the bag, which is a 3 but before grabbing the second ball, we put the 1st ball back inside the bag. Then, we grab the second ball. Q: Are the events in part (a) independent or dependent? Part (b)? Explain.

A --> dependent = 2nd ball's chance changes because the 1st ball isn't replaced. B --> independent = 2nd ball's chance is the same as the first ball.

Example 4: In 2005, 19.1% of all murder victims were between the ages of 20 and 24 years old. Also in 2005, 16.6% of all murder victims were 20-24-year-old males. What is the probability that a randomly selected murder victim in 2005 was male given that the victim is 20-24 years old?

A = ages btwn 20-24 B = victim is male P(A) = 19.1% = 0.191 P(A and B) = 16.6% = 0.166 P(B|A) = P(A and B)/P(A) = 0.166/191 = 0.8691

Example 1: Suppose our probability experiment consists of rolling a single fair die. c) Define the event A = rolling a number greater than 3

A = {4, 5, 6}

A or B symbol

A U B

A and B symbol

A ∩ B

Probability Experiment

Any process that can be repeated in which the results are certain

Law of Large Numbers

As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome (??)

Practice Law of Large Numbers: Suppose that a survey is conducted in which 500 families with three children are asked to disclose the gender of their children. Based on the results of the survey, it was found that 180 families had two boys and one girl. (b) Compute the probability of having two boys and one girl in a three-child family using the classical method, assuming having a boy and having a girl are equally likely.

B < B G G < B G --- continue from there to find sample space P(2 boys & 1 girl) = 3/8 = 0.375 --> pretty darn good prediction

Example 1: Determine whether or not the following events are independent or dependent (b) E: You are late for school F: Your car runs out of gas

Dependent E ⊥̸F

Example 1: Determine whether or not the following events are independent or dependent (d) E: Earned a Doctorate degree F: Earning more than $100,000 a year after graduation

Dependent E ⊥̸F (may not be true all the time but we can assume in general)

Example 1: Determine whether or not the following events are independent or dependent (c) E: Flipping a "heads" on a coin F: Rolling a 5 on a die

E ⫫ F

if E and F are independent (symbol??)

E ⫫ F

Example 2: Suppose our experiment is rolling a fair die (c) Find the Ec if E is rolling at least a 3

Ec = {1, 2}

Example 2: Suppose our experiment is rolling a fair die (a) Find the Ec if E = {2, 5}

Ec = {1, 3, 4, 6}

Example 2: Suppose our experiment is rolling a fair die (b) Find the Ec if E is the event of rolling an odd number

Ec = {2, 4, 6}

mutually exclusive

Events that cannot occur at the same time.

The General Addition Rule --> explain how it's different from the Addition Rule for Disjoint Events

For any two events E and F P(E or F) = P(E) + P(F) - P(E ∩ F) --> subtract so values aren't double counted!!!!

Addition Rule for Disjoint Events

If E and F are disjoint (or mutually exclusive) events, then P(E or F) = P(E) + P(F)

Example of event

If our experiment is drawing a card out of a deck, then some examples of events would be drawing a red card, drawing a face card, or drawing a red face card.

Example 1: Determine whether or not the following events are independent or dependent (a) E: The battery in your cell phone is dead F: The batteries in your calculator are dead.

Independent but not mutually exclusive (can happen at the same time).

Practice Law of Large Numbers: Suppose that a survey is conducted in which 500 families with three children are asked to disclose the gender of their children. Based on the results of the survey, it was found that 180 families had two boys and one girl. (a) Estimate the probability of having two boys and one girl in a three-child family using the empirical method.

P(2 boys & 1 girl) = 180/500 = .36 = 36%

Example 1: Suppose we are drawing a single card from a standard 52 card deck (a) Find the probability that we draw a 4 or a jack

P(4 or Jack) = 8/52 = 2/13

P(4 or more rooms) = 0.896

P(4 to 6) = P(4) + P(5) + P(6) = 0.617

General Multiplication Rule

P(A and B) = P(A) • P(B|A)

The probability that two events A and B both occur is

P(A and B) = P(A) • P(B|A)

Complement Rule of Probability

P(A) + P(Ac) = 1 OR P(A) = 1 - P(Ac) OR P(Ac) = 1 - P(A)

P(A U B) =

P(A) + P(B)

How to Calculate Conditional Probability

P(B|A) = P(A∩B) / P(A) --> B given A

Example 6: Suppose you just purchased a digital music player and have put 6 tracks on it. After listening to them you decide that you like 4 of the songs. With the random feature on your player, each of the 6 songs is played once in random order. Find the probability that among the first two songs played. b) You like neither of them.

P(D1 D2) = 2/6 • 1/5 = 2/30 = 0.07

P(D1 L2) OR P(L1 D2) (2/6 • 3/5) + (4/6 • 1/5) = 8/15 = 0.53

Example 6: Researchers found that 27% of adults 27-32-year-old say that they have danced in public while under the influence of alcohol. Randomly selected three 27-32-year-olds. • What is the probability that all three selected have not danced under the influence?

P(Dc • Dc • Dc) = (0.73)(0.73)(0.73) = 0.389

If E and F are independent events, then

P(E and F) = P(E) • P(F)

Multiplication Rule for Independent Events

P(E and F) = P(E) • P(F)

One way to determine if E and F are independent

P(E and F) = P(E) • P(F)

Probability of an Event

P(E) --> the likelihood of the event occurring

If computing probability using the Classical Method: - So if S is the sample space of this experiment

P(E) = N(E) / N(S) = # of outcomes in E / # of outcomes in S

If computing probability using the Classical Method: - If an experiment has a total of n equally likely outcomes and if the number of ways that an event E can occur is m, then the probability of E, P(E) is

P(E) = m / n = # of successes in E / total # of outcomes = want / have in total

Example 4: In a 2011 survey, students were asked how many times they checked Facebook per day. The results of the survey are listed below. Sometimes (0-3) Regularly (4-6) Frequently(7+) High School Student 782 1,302 1,356 College Student 1,234 982 867 Graduate Student 1,429 528 452 (b) What is the probability that a randomly selected student checks Facebook frequently and is a college student

P(F and C) = 867/8932 = 0.0971

Example 4: In a 2011 survey, students were asked how many times they checked Facebook per day. The results of the survey are listed below. Sometimes (0-3) Regularly (4-6) Frequently(7+) High School Student 782 1,302 1,356 College Student 1,234 982 867 Graduate Student 1,429 528 452 (e) What is the probability that a randomly selected student checks Facebook frequently and is a graduate student?

P(F and G) = 452/8932 = 0.0506

Example 5: Failure of an airplane part can result in catastrophe. Many airline parts utilize "triple modular redundancy." Each part has two back-up parts in case the initial part fails. Suppose the probability of failure of a certain part is 0.006 • Assuming each part's success/failure is independent of the others, what is the probability that all three parts fail resulting in disaster?

P(F, F, F) = P(F) • P(F) • P(F) = (0.006)(0.006)(0.006) = 2.2 x 10^-7

Example 8: Suppose G and H are two events and that P(G) = 0.3, P(H) = 0.2, and P(G and H) = 0.06 Are G and H independent?

P(G and H) = P(G) • P(F) 0.06 = (0.3)(0.2) --> A and B are independent

P(L1 L2) = (4/6) • (3/5) = 12/30 = 0.4 0.4 --> this is big # --> not unusual

Example 4: In a 2011 survey, students were asked how many times they checked Facebook per day. The results of the survey are listed below. Sometimes (0-3) Regularly (4-6) Frequently(7+) High School Student 782 1,302 1,356 College Student 1,234 982 867 Graduate Student 1,429 528 452 (c) What is the probability that a randomly selected student checks Facebook regularly or is a high school student?

P(R or HS) --> P(R) + P(HS) - P(R ∩ HS) = 0.5542

Example 4: In a 2011 survey, students were asked how many times they checked Facebook per day. The results of the survey are listed below. Sometimes (0-3) Regularly (4-6) Frequently(7+) High School Student 782 1,302 1,356 College Student 1,234 982 867 Graduate Student 1,429 528 452 (a) What is the probability that a randomly selected student checks Facebook regularly

P(R) = 2812/8932 = 0.3148

Example 4: In a 2011 survey, students were asked how many times they checked Facebook per day. The results of the survey are listed below. Sometimes (0-3) Regularly (4-6) Frequently(7+) High School Student 782 1,302 1,356 College Student 1,234 982 867 Graduate Student 1,429 528 452 (d) What is the probability that a randomly selected student checks Facebook sometimes or is a graduate student?

P(S OR G) --> P(S) + P(G) - P(S ∩ G) = 0.4954

Example 4: Suppose an experiment is drawing a card from a standard 52 card deck. (c) Find the probability of drawing a two

P(a 2) = 4/52 = 1/13

Example 1: Suppose we are drawing a single card from a standard 52 card deck (c) Find the probability that we draw an ace, kind, or queen

P(ace or king or queen) = 12/52 = 3/13

P(at least 2) = 1 - P(1) = 1 - 0.005 = 0.995

Example 5: Failure of an airplane part can result in catastrophe. Many airline parts utilize "triple modular redundancy." Each part has two back-up parts in case the initial part fails. Suppose the probability of failure of a certain part is 0.006 • What is the probability that at least one of the parts does not fail?

P(at least one doesn't fail) = 1 - P(all 3 parts fail) = 1 - P(F, F, F) = 1 - (2.2 x 10^-7) = 0.99999978

Example 6: Researchers found that 27% of adults 27-32-year-old say that they have danced in public while under the influence of alcohol. Randomly selected three 27-32-year-olds. • What is the probability that at least one of the three selected has not danced under the influence?

P(at least one has NOT) = 1 - P(all has danced) = 1 - P(D • D • D) = 1 - (0.27)(0.27)(0.27) = 0.9803

Example 5: You receive a shipment of 12 TVs. Three are defective. b) If you randomly choose two TVs, what is the probability that at least one does not work?

P(at least one is D) = 1 - P(None is D) = 1 - P(G1 G2) = 1 - 0.545 = 0.455

The two equations for At least

P(at least one is) = 1 - P(all are not) P(at least one is not) = 1 - P(all are)

Example 1: Suppose we are drawing a single card from a standard 52 card deck (b) Find the probability that we draw a black card or a red card

P(black card or a red card) = 1

Example 4: Suppose an experiment is drawing a card from a standard 52 card deck. (b) Find the probability of drawing a black card

P(black card) = 26/52 = 1/2

Example 4: Suppose an experiment is drawing a card from a standard 52 card deck. (d) Find the probability of drawing a black face card

P(black face card) = 6/52 = 3/26

Example 1: Suppose we are drawing a single card from a standard 52 card deck (g) Find the probability that a black card or a diamond is drawn

P(black or diamond) = 26 + 13/ 52 = 39/52 = 3/4

Example 1: Suppose we are drawing a single card from a standard 52 card deck (d) Find the probability that we draw a heart or an ace

P(heart or an ace) = 13 + 4 - 1 /52 = 16/52 = 4/13

Example 4: Suppose an experiment is drawing a card from a standard 52 card deck. (a) Find the probability of drawing a heart

P(heart) = 13/52 = 1/4

P(less than 8) = 1 - P(8 or more) = 1 - 0.156 = 0.844

Example 6: In a national survey conducted by the Centers of Disease Control in order to determine college students' health risk behaviors, college students were asked "How often do you wear a seatbelt with a car?" The frequencies were recorded below: RESPONSE Frequency Never 118 Rarely 249 Sometimes 345 Most of the Time 716 Always 3,093 (a) Approximate the probability that a randomly selected college student never wears a seatbelt in the car.

P(never wears a seatbelt) = 118/4521 = 0.026 = 3%

Example 1: Suppose we are drawing a single card from a standard 52 card deck (f) Find the probability that a red card or a diamond is drawn

P(red or diamond) = 26 + 13 - 13 /52 = 26/52 = 1/2

Example 2: Suppose we have a bag with 4 balls, all of which are numbered 1 through 4 and all of which are equally likely to be pulled out of the bag. In our experiment, we wish to pull two balls out of the bag, and we want to know what the probability will be if the second ball we draw out has a number 2. (a) Suppose we grab the 1st ball out of the bag, which is a 3. Then, we leave that ball out of the bag and grab the second ball. What is the probability that the second ball is a 2? --> What is this called?

P(second ball is 2) = 1/3 --> called sampling w/o replacement

Example 2: Suppose we have a bag with 4 balls, all of which are numbered 1 through 4 and all of which are equally likely to be pulled out of the bag. In our experiment, we wish to pull two balls out of the bag, and we want to know what the probability will be if the second ball we draw out has a number 2. (b) Suppose we grab the 1st ball out of the bag, which is a 3 but before grabbing the second ball, we put the 1st ball back inside the bag. Then, we grab the second ball. What is the probability that the second ball is a 2? --> What is this called?

P(second ball is 2) = 1/4 --> called sampling with replacement

Example 1: Suppose our probability experiment consists of rolling a single fair die. a) Identify the sample space

S = {1, 2, 3, 4, 5, 6, 7}

Rolling a die and define A = getting an even number - Find Sample Space - Find A = - Find Ac =

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

Suppose a woman has two children. What is the sample space for the genders of the two children?

S = {BB, BG, GG, BB} 2 • 2 = 4 outcomes [2 possibilities for 1st child (B, G)] • [2 possibilities for 1st child (B, G)]

If our experiment is flipping a coin, then the sample space is

S = {head, tail}

Example of mutually exclusive

Suppose we are rolling a fair die and let A be the event that we roll an odd and let B be the event that we roll an even. Then, we can say events A and B are disjoint or mutually exclusive events.

Conditional Probability

The notation P(B|A) is read "the probability of event B given event A." It is the probability that the event B occurs, given that event A has already occurred.

Approximating Probabilities Using the Empirical Approach

The probability of an event E is approximately the number of times event E is observed divided by the number of repetitions of the experiment OR P(E) ≈ relative frequency of E = frequency of E/number of trials of experiment

to determine the total number of outcomes use the

counting principle (multiply the outcomes of each individual task)

Conditional Probability and Independence

Two events K and F are independent if P(F | K) = P(F) or P(K and F) = P(K) • P(F)

counting principle

a simple way to find the number of outcomes (# of outcomes for task 1) • (# of outcomes for task 2) • ...

Example 1: Suppose that a single six-sided die is rolled. a) What is the probability that the die comes up 4? b) Now suppose that the die is rolled a second time, but we are told the outcome will be an even number. What is the probability that the die comes up 4?

a) P(4) = 1/6 b) P(4) = 1/3

Example 3: A survey was conducted by the Gallup Organization in which 1,017 adult Americans were asked, "Which of the following statements comes closest to your belief about God?" The results of the survey, by region of the country, are given in the table Believe in God Believe in universal spirit Don't believe East 204 36 15 Midwest 212 29 13 South 219 26 9 West 152 76 26 a) What is the probability that a randomly selected adult American who lives in the East believes in God? b) What is the probability that a randomly selected adult American who believes in God lives in the east? --> what does this show?

a) P(God | East) = 204/255 = 0.8 b) P(East | God) = 204/787 = 0.26 shows --> P(B|A) ≠ P(A|B)

Example 2: Suppose we are drawing a single card from a standard 52 card deck. a) What is the probability that the card that is drawn is a heart? b) Now suppose we are still drawing a single card, but we know that the card is red. What is the probability that the card drawn is a heart?

a) P(H) = 13/52 = 1/4 b) P(H) = 13/26 = 1/2

can usually use the _______ to find the probabilities of the form "at least one"

complement rule --> sum of the probabilities of an event and its complement must equal 1

OR =

add

Event

any collection of outcomes from a probability experiment.

To determine whether two events are indp

ask yourself whether the probability of one event is affected by the other event

Suppose we roll a die, and let A = even numbers, B = getting a number less than 4 --> what's the sample space? --> what does A ∩ B = --> Ac = --> Bc =

check page 6 in chapter 5 notes A ∩ B = {2} Ac = {1, 3, 5} Bc = {4, 5, 6}

Suppose we roll a die, and let A = even numbers, B = getting a number less than 4 --> what's the sample space? --> what does A U B = ?

check page 6 of chapter 5 lecture notes A U B = {1, 2, 3, 4, 6}

Rules of Probabilities 4. An unusual event is an event that has a low probability of occurring. Typically, an event with a probability _______ is considered unusual.

closer to 0

sampling without replacement

dependent

Example 5: Suppose the experiment is rolling a pair of fair dice (d) Find the probability that the sum is divisible by 4

divisible of 4 --> multiple of 4 sum = 4 <-- 3 | (1, 3) (2, 2) (3, 1) sum = 8 <-- 5 | (2, 6) (3, 5) (4, 4) (5, 3) (6, 2) sum = 12 <-- 1 | (6, 6) P (sum is divisible by 4) = 9/36 = 1/4

And

find

examples of an experiment

flip a coin, roll a die, draw a card from a deck of cards

sampling with replacement

independent (won't matter what you chose first)

A or B

is the event consisting of elements in A or in B or in both A and B

A and B

is the event consisting of elements that are in both A and B

Complement of A

is the event consisting of elements that are in the sample space that are not in A.

At least =

means greater than or equal to

Probability

measures the likelihood of an event occurring

Another name for disjoint events are

mutually exclusive events

Complement Rule

n(A) + n(Ac) = n(S) # of elements in A + # of elements in Ac = total # of elements in sample space

Are the two events A = grabbing a red marble and B = grabbing a green marble equally likely if we are grabbing from a bag of 3 red marbles and 2 green marbles.

no, more red than green

A and B in other words

the intersection of A and B

Two events, E and F are independent if

the occurrence of event E in an experiment does not affect the probability of event F.

Two events are dependent if

the occurrence of event E in an experiment will affect the probability of event F occurring

A or B in other words

the union of A with B

Complement of A in babie terms

the values in the sample space that aren't in A

Two events are disjoint if...

they have no outcomes in common

empirical method in babie

using actual data to predict

Example 3: Determine whether each experiment has equally likely outcomes. (a) Picking a card out of a deck

Chapter 5 Stats

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