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}