Chapter 4 - Probability and Probability Distributions

Question 1

As n gets larger… Sample -> ? Relative Frequence -> ?

Answer 1

Sample -> Population Relative Frequency -> Probability

Question 2

The process by which an observation (or measurement) is obtained

Answer 2

Experiment

Question 3

The outcome that is observed on a single repetition of the experiment

Answer 3

Simple Event

Question 4

A Simple Event is denoted by?

Answer 4

E with a subscript

Question 5

Each simple event will be assigned a __ measuring __ ___ it occurs.

Answer 5

Probability How Often (How Likely)

Question 6

The set of all simple events of an experiment

Answer 6

Sample Space, S

Question 7

A collection of one or more simple events (a subset of sample space)

Answer 7

an Event

Question 8

Two events are __ __ if, when one event occurs, the other cannot, and vice versa.

Answer 8

mutually exclusive

Question 9

The probability of an event A measures “how often” we think A will occur. We write?

Answer 9

P(A)

Question 10

If we let n get infinitely large

Answer 10

P(A) = lim(n->∞) f/n

Question 11

P(A) must be between _ and _

Answer 11

0 and 1

Question 12

The sum of the probabilities for all simple events in S equals __.

Answer 12

1

Question 13

The probability of an event A is found by..

Answer 13

adding the probabilities of all the simple events contained in A.

Question 14

If the simple events in an experiment are equally likely, you can calculate… You can use ___ ___ to find n(a) and N

Answer 14

P(A) = n(a) / N Counting Rules

Question 15

If an experiment is performed in two stages, with m ways to accomplish the first stage and n ways to accomplish the second stage, then…

Answer 15

There are mn ways to accomplish the experiment

Question 16

The mn Rule is easily extended to k stages with the number of ways equal to..

Answer 16

n(1)n(2)n(3)…n(k)

Question 17

The number of ways you can arrange
n distinct objects, taking them r at a time

Answer 17

Permutations

Question 18

The number of distinct combinations of n distinct objects that can be formed, taking them r at a time

Answer 18

Combinations

Question 19

A ∪ B

Answer 19

The UNION of two events, A and B

Question 20

The Union of two events, A and B is the event that..

We write…

Answer 20

Either A or B or BOTH occur when the experiment is performed.

A ∪ B

Question 21

A ∩ B

Answer 21

the Intersection of two events, A and B

Question 22

the Intersection of two events, A and B

We write..

Answer 22

Is the event that both A and B occur when the experiment is performed

A ∩ B

Question 23

If two events A and B are mutually exclusive then P(A ∩ B) = ?

Answer 23

P(A ∩ B) = 0

Question 24

What consists of all outcomes of the experiment that do not result in event A?

We write?

Answer 24

The Complement of an event A

A^c

Question 25

The Additive Rule for Unions

For any two events, A and B, the probability of their union P(A∪B) is

Answer 25

P(A∪B) = P(A) +P(B) - P(A∩B)

Question 26

When two events A and B are mutually exclusive, P(A∩B) = 0
and?

Answer 26

P(A∪B) = P(A) + P(B)

Question 27

P(A ∩ A^c) =

Answer 27

0

Question 28

P(A ∪ A^c) =

Answer 28

1

Question 29

P(A^c) =

Answer 29

1-P(A)

Question 30

The rule for calculating P(A  B) depends on the idea of ___ and ___ events

Answer 30

Independant, Dependant

Question 31

Two events, A and B, are said to be independent if and only if

Answer 31

the probability that event A occurs does not change, depending on whether or not event B has occurred.

Question 32

The probability that A occurs, given that event B has occurred is called the ___ ___ of A given B

Answer 32

conditional probability

Question 33

Conditional Probability of A given B is defined as

Answer 33

Question 34

P(A | B)

the line seperating A and B means

Answer 34

“given”

Question 35

Two events A and B are independent if and only if

Answer 35

P(A|B) = P(A)

AND

P(A∩B) = P(A)P(B)

Question 36

The Multiplicative Rule for Intersections

For any two events, A and B, the probability that both A and B occur is…

Answer 36

P(A∩B) = P(A) P(B|A)

        = P(B) P(A|B)

Question 37

If the events A and B are independent, then the probability that both A and B occur is

Answer 37

P(A∩B) = P(A) P(B)

Question 38

Law of Total Probabilty

Let S1, S2, S3,…,Sk be mutually exclusive. Then the probability of another event A can be written as..

Answer 38

P(A) = P(A∩S1) + P(A∩S2) +…+P(A∩Sk)

= P(S1)P(A|S1) +P(S2)P(A|S2) +…+P(Sk)(P(A|Sk)

Question 39

Bayes Rule

Let S1 , S2 , S3 ,…, Sk be mutually exclusive and exhaustive events with prior probabilities P(S1), P(S2),…,P(Sk). If an event A occurs, the posterior probability of Si, given that A occurred is

Answer 39

Question 40

A quantitative variable x is a __ ___ if the value that it assumes,

corresponding to the outcome of an experiment is a chance or random event.

Answer 40

Random Variable

Question 41

Random variables can be __ or ___.

Answer 41

discrete or continuous

Question 42

The probability distribution for a discrete random variable x is a graph, table or formula that gives the ..

Answer 42

possible values of x and the probability p(x) associated with each value.

Question 43

Probability distributions can be used to describe the population

(3)

Answer 43

Shape

Outliers

Center and Spread (mean and standard deviation)

Question 44

Let x be a discrete random variable with probability distribution p(x).

Then the mean, variance and standard deviation of x are given as

Answer 44

Mean: µ = Σx p(x)

Variance: σ² = Σ(x-µ)² p(x)

Standard Deviation: σ = √ σ²