Exam 2

Question 1

Assigning Probabilities: Classical Approach

Answer 1

Assigning probabilities based on the assumption of equally likely outcomes

Question 2

Assigning Probabilities: Relative Frequency Approach

Answer 2

Assigning probabilities based on experimentation or historical data

Question 3

Assigning Probabilities: Subjective Approach

Answer 3

Assigning probabilities based on judgment

Question 4

Classical Approach example

Answer 4

If an experiment has n possible outcomes, this method would assign a probability of 1/n to each outcome. It is necessary to determine the number of possible outcomes. Experiment: Rolling a die Outcomes {1, 2, 3, 4, 5, 6} Probabilities: Each sample point has a 1/6 chance of occurring

Question 5

Relative Frequency Method

Answer 5

Each probability assignment is given by dividing the frequency (number of days) by the total frequency (total number of days).

Question 6

Subjective Approach

Answer 6

“In the subjective approach we define probability as the degree of belief that we hold in the occurrence of an event”

Example: Weather forecasting
it’s a subjective probability based on past observations combined with current weather conditions

– based on current conditions, there is a 60% chance of rain (say)

Question 7

Compliment of an Event

Answer 7

The complement of event A is defined to be the event consisting of all sample points that are “not in A”.

Complement of A is denoted by Ac

The Venn diagram below illustrates the concept of a complement.

P(A) + P(Ac ) = 1

Question 8

Intersection of events

Answer 8

The intersection of events A and B is the set of all sample points that are in both A and B.

The intersection is denoted: A and B A ∩ Β

The joint probability of
A and B is the probability of
the intersection of A and B
P(A and B)

Question 9

Union of two events

Answer 9

The union of two events A and B, is the event containing all sample points that are in A or B or both:

Union of A and B is denoted: A or B (A ∪ B)

Question 10

Mutually Exclusive Events

Answer 10

When two events are mutually exclusive (that is the two events cannot occur together), their joint probability is 0, hence:

Mutually exclusive: no points in common
For example A = tosses totaling 7 and B = tosses totaling 11

Question 11

Condtional Probability

Answer 11

Conditional probability is used to determine how two events are related; that is, we can determine the probability of one event given the occurrence of another related event.

Conditional probabilities are written as P(A | B) and read as “the probability of A given B” and is calculated as:

Question 12

Independence

Answer 12

If both events can occur they are dependent.

One of the objectives of calculating conditional probability is to determine whether two events are related.

In particular, we would like to know whether they are independent, that is, if the probability of one event is not affected by the occurrence of the other event.

Two events A and B are said to be independent if
P(A|B) = P(A)
or
P(B|A) = P(B)

Question 13

Three probability rules and trees

Answer 13

3 rules that enable us to calculate the probability of more complex events from the probability of simpler events:

The Complement Rule

The Multiplication Rule

The Addition Rule

Question 14

Probability Rule: Complement Rule

Answer 14

The complement rule gives us the probability of an event NOT occurring

P(AC) = 1 – P(A)

Example:

In the simple roll of a die, the probability of the number “1” being rolled is 1/6
The probability that some number other than “1” will be rolled is 1 – 1/6 = 5/6.

Question 15

Probability Rule: Multiplication Rule

Answer 15

The multiplication rule is used to calculate the joint probability of two events. It is based on the formula for conditional probability defined earlier:

If we multiply both sides of the equation by P(B) we have:

P(A and B) = P(A | B)•P(B)

Likewise, P(A and B) = P(B | A) • P(A)

If A and B are independent events, then P(A and B) = P(A)•P(B)

Question 16

Probability Rule: Addition Rule

Answer 16

Recall: the addition rule is used to compute the probability of event A or B or both A and B occurring; i.e. the union of A and B.

Question 17

Random Variable

Answer 17

A random variable is a function or rule that assigns a number to each outcome of an experiment.

Question 18

Discrete Random Variable

Answer 18

Discrete Random Variable
– one that takes on a countable number of values
– E.g. values on the roll of dice: 2, 3, 4, …, 12

Question 19

Continuous Random Variable

Answer 19

Continuous Random Variable
– one whose values are not discrete, not countable
– E.g. time (30.1 minutes? 30.10000001 minutes?)

Question 20

Binomial Distribution

Answer 20

The binomial distribution is the probability distribution that
results from doing a “binomial experiment”.
Binomial experiments have the following properties:

Fixed number of trials, represented as n.
Each trial has two possible outcomes, a “success” and a “failure”.
P(success)=p (and thus: P(failure)=(1–p), for all trials.

The trials are independent, which means that the outcome of one trial does not affect the outcomes of any other trials.

Question 21

Conditions of Binomial Expierement:

Answer 21

 There is a fixed finite number of trials (n=10).
An answer can be either correct or incorrect.
The probability of a correct answer
(P(success)=.20) does not change from
question to question.
 Each answer is independent of the others.

Question 22

Poisson Random Variable

Answer 22

The Poisson random variable is the number of successes that occur in a period of time or an interval of space in a Poisson experiment.

E.g. On average, 96 (successes) trucks arrive at a border crossing
 every hour (time period).

E.g. The number of typographic errors in a new textbook edition averages 1.5 per 100 pages.

Question 23

Binomial Distribution Example

Answer 23

The quiz consists of 10 multiple-choice questions. Each question has five possible answers, only one of which is correct. Pat plans to guess the answer to each question.

Algebraically then: n=10, and P(success) = 1/5 = .20

Question 24

Poisson Distribution

Answer 24

A statistics instructor has observed that the number of typographical errors in new editions of textbooks varies considerably from book to book. After some analysis he concludes that the number of errors is Poisson distributed with a mean of 1.5 per 100 pages. The instructor randomly selects 100 pages of a new book. What is the probability that there are no typos?

Question 25

Probability Tree

Answer 25

Question 26

Normal Distribution

Answer 26

The normal distribution is the most important of all probability distributions. The probability density function of a normal random variable is given by:

Question 27

The normal distribution is described by two parameters

Answer 27

its mean μ and its standard deviation σ . Increasing the mean shifts the curve to the right…

Question 28

Second Parameter

Answer 28

Increasing the standard deviation (σ) “flattens” the curve…

Question 29

Normal Probablity Z equation

Answer 29

Question 30

Standard Normal Distribution

Answer 30

A normal distribution whose mean is zero and standard deviation is one is called the standard normal distribution.

Question 31

Sampling Distributions

Answer 31

A sampling distribution is created by, as the name suggests, sampling.

One of the ways to create a sampling distribution is to rely on rules of probability

For example, consider the roll of one and two dice…

Question 32

Sampling Distribution of the Mean equation

Answer 32

Question 33

Central Limit Theorum

Answer 33

The sampling distribution of the mean of a random sample drawn from any population is approximately normal for a sufficiently large sample size.

The larger the sample size, the more closely the sampling distribution of X will resemble a normal distribution.

In most practical situations, a sample size of 30 may be
sufficiently large to allow us to use the normal
distribution as an approximation for the sampling
distribution of X.

Question 34

Sampling Distribution of the Sample Mean

Answer 34

mu xbar = mu

σ²xbar = σ²/n and σ xbar = σ / sq rt of n

3. If X is normal, xbar is normal. If X is non-normal, xbar is approximately normal for sufficiently large sample sizes.

Question 35

The standard deviation of the sampling distribution is called the ________ _______

Answer 35

Standard Error

Question 36

Using the Sampling Distribution for Inference

Answer 36

Question 37

Normal approximation to the binomial works best when the number of experiment n (sample size) is_______

Answer 37

large

Question 38

For the approximation to provide good results two conditions should be met:

Answer 38


1) np ≥ 5
2) n(1–p) ≥ 5

Question 39

Normal approximation to Binomial equations

Answer 39

mean

variance

standard deviation

Question 40

Sampling Distribution of a Proportion

Answer 40

The estimator of a population proportion of successes is the sample proportion. That is, we count the number of successes in a sample and compute:

Question 41

Sampling Distribution of a Proportion Z score equation

Answer 41

Question 42

Marginal Probability

Answer 42

Marginal probability is the probability of the occurrence of the single event

Question 43

Joint Probability

Answer 43

Joint probability is the probability that two events will occur simultaneously.

Question 44

Requirements of Probability

Answer 44

(1) That the probabilities lie between 0 and 1.
(2) The sum of all probabilities of the distribution sum up to 1.

Question 45

negative z value

Answer 45

to the left of the mean

Question 46

positive z score

Answer 46

to the right of the mean