Statistics - Chapter 3

Define classical (theoretical) probability

Each outcome in a sample space is equally likely

Define conditional probability

The probability of an event occurring, given that another event has already occurred.

The formula to find the number of combinations of r objects selected from a group of n objects without regard to order

nCr = n! / (n-r)! * r!

The complement of​ "at least​ one" is ________

"none"

The probability that a person in the United States has type B​+ blood is 7​%. How would you find the probability of selecting four unrelated people in the US at random and all four have type B​+ blood?

(.07) * (.07) * (.07) * (.07)

The probability that a person in the United States has type B​+ blood is 7​%. How would you find the probability of selecting four unrelated people in the US at random and none of the four have type B​+ blood?

(.93) * (.93) * (.93) * (.93)

A coin is tossed and a die is rolled, how would you find the probability of getting a head and rolling a 4 in sequence using the Multiplication Rule?

(1/2) * (1/6)

A coin is tossed and a six​-sided die is rolled. How would you find the probability of tossing a tail and then rolling a number greater than 1?

(1/2) * (5/6)

You randomly select an integer from 0 to 19 ​(inclusively) and then randomly select an integer from 0 to 4 ​(inclusively). How would you calculate the probability of selecting a 2 both times?

(1/20) + (1/5)

You select a random card from a deck. How to calculate the probability of the card being a seven or a spade?

(4/52) + (13/52) - (1/52)

You select a random card from a deck. How to calculate the probability of the card being a seven or an eight?

(4/52) + (4/52)

You select a random card from a deck. How to calculate the probability of the card being a seven, eight, or jack?

(4/52) + (4/52) + (4/52)

How to simplify 5!/3!

(5*4) * (3!) / 3! 3! cancels 5 * 4 = 20

How to calculate the empirical (or statistical) probability of event E. P(E) =

(frequency of event E) / total frequency f/n where n = Σf

How to calculate the classical (theoretical) probability of event E. P(E) =

(number of outcomes in event E) / (total number of outcomes in sample space)

Which numbers could not be used to represent the probability of an event? Why? .0002 320/1058 -1.5 33.3% 64/25 0

-1.5 and 64/25 because probability can't be less than 0 or greater than 1

How would you find the number of ways 10 video games can be arranged side-by-side on a shelf?

10!

A computer is used to select a number randomly between 1 and 9 inclusive. Event C is selecting a number greater than 7. What is P(C)?

2/10 or 20% or 0.2

There are 46 runners in a race. How many ways can the runners finish​ first, second, and​ third?

46P3 46! / (46-3)!

3! expanded calculation

3 * 2 * 1 = 6

A roulette wheel consists of slots numbered 00, 0, 1, 2, ... , 31. A ball is spun and lands in one of the numbered slots. How many outcomes are in the sample space?

33

Assuming that no questions are left​ unanswered, how would you calculate how many ways a 6 question quiz with 4 multiple choice questions can be​ answered?

4 ^ 6 = 4096

Identify the number of outcomes in the sample space of the probability experiment of randomly choosing a multiple of 5 between 21 and 49

5

A restaurant offers a​ $12 dinner special that has 5 choices for an​ appetizer, 11 choices for an​ entrée, and 3 choices for a dessert. How many different meals are available when you select an​ appetizer, an​ entrée, and a​ dessert?

5 * 11 * 3

The probability that an event will happen is 17/23. Find the probability that the event will not happen (as a fraction).

6/23

Give a set of letters AAAABBC, how to find the number of distinguishable permutations of the letters?

7! / 4! * 2! * 1! = (7*6*5) / 2

How to solve: a 9-key keypad with letters A, B, C, D, E, F, G, H, and I where each key may only be used once. How many codes are possible?

9P9

Example of subjective probability

A doctor giving a patient a 90% chance of recovery given health, extent of injuries, etc. A business analyst predicts the chance employees of a company will go on strike is 0.25

Define simple event

An event that consists of exactly one outcome

What is the difference between an outcome and an event?

An outcome is the result of a single probability experiment. An event is a set of one or more possible outcomes

Define the Law of Large Numbers

As an experiment is repeated over and over, the empirical probability of an event approaches the theoretical (actual) probability of the event

Classical, Empirical, or Subjective? The probability of winning a 1000-ticket raffle with one ticket is 1/1000

Classical - you know the number of outcomes and each is equally likely

If order doesn't matter - permutation or combination?

Combination

Permutations, Combinations, or Neither? The number of ways a four-member committee can be chosen from 10 people.

Combination. The order of the committee members does not matter.

The probability that a person in the United States has type B​+ blood is 7​%. How would you find the probability of selecting four unrelated people in the US at random and at least one of the four have type B​+ blood?

Complement of at least one is none so find the probability of none and subtract it from 1. 1 - ((.93) * (.93) * (.93) * (.93))

Independent or dependent events? A: driving over 85 mph B: getting in a car accident

Dependent

Two cards are selected from a standard deck of 52 playing cards. The first card is not replaced before the second card is selected. How would you find the probability of selecting an eight and then selecting a king?

Dependent events (4/52) * (4/51)

Independent or dependent events? A: selecting a king from a standard deck of cards and not replacing it B: selecting a queen from the same deck

Dependent, P(B) changes depending on the outcome of A

Independent or dependent events? A: study 4 hours for an exam B: get an A on an exam

Dependent, P(B) changes depending on the outcome of A

How is the complement of event E denoted?

E' "E prime"

Classical, Empirical, or Subjective? According to company records, the probability that a washing machine will need repairs during a six-year period is 0.16.

Empirical, based on observations from the company records

Classical, Empirical, or Subjective? The probability that a voter chosen at random will be younger than 35 years old is 0.3

Empirical, most likely based on a survey of sample voters

T/F the probability of an event can be greater than 1 or less than 0

False

T/F two mutually exclusive events can have one outcome in common

False

T/F P(E) = 1 + P(E')

False, P(E) = 1 - P(E')

T/F an event in a probability experiment can only have one outcome

False, an event can consist of one or more outcomes

T/F You toss a fair coin nine times and it lands tails up each time. The probability it will land heads up on the tenth flip is greater than 0.5.

False, it is exactly 0.5

T/F a combination is an ordered arrangement of objects

False, that would be a permutation

T/F You toss a coin and roll a die. The event​ "tossing tails and rolling a 2 or 3​" is a simple event.

False, the event is not simple because it consists of two possible outcomes

Independent or dependent events? A: tossing a coin and getting heads B: rolling a six-sided die and getting a 6

Independent

The probability that a knee surgery is successful is 0.85. How would you find the probability that three knee surgeries in a row are successful?

Independent events (0.85) * (0.85) * (0.85)

Independent or dependent events? A: drink milk at lunch B: get an A on an exam

Independent, the occurrence of B is not impacted by the occurrence of A

If order matters - permutation or combination?

Permutation

Are these two events mutually exclusive? A: Randomly select a blood donor with type O blood B: Randomly select a female blood donor

No, you could select a female blood donor with type O blood.

Are these two events mutually exclusive? A: randomly select a male student B: randomly select a nursing major

No, you could select a male nursing major which is a possible outcome of both events.

Two events A and B are independent, what is the probability that A and B will occur in sequence. P (A and B) =

P(A and B) = P(A) * P(B)

Two events A and B are dependent, what is the probability that A and B will occur in sequence. P (A and B) =

P(A and B) = P(A) * P(B|A)

Difference between P(A | B) and P(A and B)?

P(A | B) is the probability that event A will occur given that B has already occurred. P(A and B) is the probability that events A and B will occur in sequence.

P(A) = .04 P(B) = .05 P(A and B) = .01 How to find P(A or B)?

P(A) + P(B) - P(A and B) .04 + .05 - .01

Permutations, Combinations, or Neither? How many different 4-letter passwords can be formed from the letters A, B, C, D, E, F, and G if no repetition of letters is allowed?

Permutation, the order in which the letters are selected matters

Permutations, Combinations, or Neither? Number of ways 11 people can line up in a row for concert tickets.

Permutation. The order matters.

What is the term for an event that consists of a single outcome?

Simple event

Classical, Empirical, or Subjective? The probability you will get an A on your next test is 0.9

Subjective, based on an educated guess

Classical, Empirical, or Subjective? The probability you will get the flu this year is 0.1

Subjective, based on an educated guess

Explain how the complement can be used to find the probability of getting at least one item of a particular type.

The complement of​ "at least​ one" is​ "none." P(at least one) = 1 − ​P(none)

When you calculate the number of permutations of n distinct objects taken r at a​ time, what are you​ counting?

The number of ordered arrangements of n objects taken r at a time.

Define dependent events

The occurrence of one of the events affects the probability of the occurrence of the other event.

Define independent events

The occurrence of one of the events does not affect the probability of the occurrence of the other event

P(B | A) in words

The probability of B, given A. The conditional probability of event B occurring, given that event A has already occurred.

Define the Range of Probabilities Rule

The probability of an event E is between 0 and 1, inclusive. 0 ≤ P(E) ≤ 1

Say in words: P(E)

The probability of event E

Define the Addition Rule if A and B are mutually exclusive

The probability that events A or B will occur is given by P(A or B) = P(A) + P(B)

Define the Addition Rule if A and B are not mutually exclusive

The probability that events A or B will occur is given by P(A or B) = P(A) + P(B) - (P(A and B))

P(A and B) in words

The probability that two events A and B will occur in sequence

Define the Multiplication Rule

The probability that two events A and B will occur in sequence is P(A and B) = P(A) * P(B|A) if A and B are dependent P(A and B) = P(A) * P(B) if A and B are independent

T/F 7C5 = 7C2

True

T/F P(E) + P(E') = 1

True

T/F Probabilities can be written as fractions, decimals, or percents

True

T/F When an event is almost certain to​ happen, its complement will be an unusual event.

True

T/F the number of different ordered arrangements if n distinct objects is n!

True

T/F the probability of an event can be 0 or 1

True

Define mutually exclusive events

Two events A and B cannot occur at the same time. A and B have no outcomes in common.

What is the difference between independent and dependent​ events?

Two events are independent when the occurrence of one event does not affect the probability of the occurrence of the other event. Two events are dependent when the occurrence of one event affects the probability of the occurrence of the other event.

Are these two events mutually exclusive? A: Roll a 3 on a die B: Roll a 4 on a die

Yes, each event only has one outcome. These two events can't occur at the same time so A and B are mutually exclusive.

A computer is used to select a number randomly between 1 and 9 inclusive. Event A is selecting a number greater than 8. Is A a simple event?

Yes, event A has exactly one outcome.

Define event

a subset of the sample space - may consist of one or more outcomes

Define probability experiment

an action, or trial, through which specific results (counts, measure, or responses) are obtained.

Define permutation

an ordered arrangement of objects. The number of different permutations of n distinct objects is n! "n factorial"

Define empirical (or statistical) probability

based on observations obtained from probability experiments

If two events are mutually​ exclusive, why is P(A and B) = 0​?

because A and B cannot occur at the same time

What are the three types of probability?

classical (theoretical) probability, empirical (statistical) probability, and subjective probability

Define the Fundamental Counting Principle

if one event can occur in m ways and a second event can occur in n ways, the number of ways the two events can occur in sequence is m * n

How to denote the number of different permutations of n distinct objects?

n! n factorial

What is the formula for finding the number of distinguishable permutations of n objects, where n1 are of one type and n2 are of another type, etc.?

n! / n1! * n2! * ... * nk! where n1 + n2 + ... + nk = n

The number of permutations of n distinct objects taken r at a time is:

nPr = (n!) / (n - r)! where r ≤ n

Each year 33 cars start the Indianapolis 500. How many ways can the cars finish first, second, and third?

nPr = 33P3 33! / (33-3)!

How to find the number of ways to form four digit codes in which no digit is repeated?

nPr =10P4 = 10! / (10-4) ! = 10! / 6! = 10 * 9 * 8 * 7 = 5040

Define subjective probability

result from intuition, educated guesses, and estimates

The number of ways to choose r objects from n objects without regard to order is called:

the number of combinations of ne objects taken r at a time

Define outcome

the result of a single trial in a probability experiment

Define the complement of event E

the set of all outcomes in a sample space that are not included in event E

Define sample space

the set of all possible outcomes of a probability experiment

When do you use the Multiplication Rule?

to find the probability of two events occurring in sequence

A roulette wheel consists of slots numbered 00, 0, 1, 2, ... , 31. A ball is spun and lands in one of the numbered slots. What is the sample space?

{00, 0, 1, 2, ... , 31}

Identify the sample space of the probability experiment of randomly choosing a multiple of 5 between 21 and 49

{25, 30, 35, 40, 45}

A computer is used to select a number randomly between 1 and 9 inclusive. Event B is selecting a number greater than 7. What is the sample space?

{8, 9}