Chapter 6: Probability

Question 1

Random experiment

Answer 1

An action or process that leads to one of several possible outcomes

• need exhaustive list of mutually exclusive outcomes

Question 2

Sample space

Answer 2

The sample space of a random experiment is a list of all possible outcomes of that experiment.

List must be exhaustive and mutually exclusive

Use set notation to represent all outcomes

S={. }

Question 3

Requirements of probabilities

Answer 3

Given a sample space the probabilities assigned to outcomes must satisfy two requirements

Where P(Oi) is the probability of outcome i

1) probability of any outcome must lie between 0 and 1

0>= P(Oi) <=1

2) the sum of the probabilities of all the outcomes in an sample space must be 1

Question 4

Classical approach of assigning probabilities

Answer 4

When answer is equally possible/ pure chance the probability of each outcome is 1/Number of outcomes

Question 5

Relative frequency approach for assigning probabilities

Answer 5

Probability= long run frequency with which an outcome occurs

Samples to determine relative probabilities (history of outcomes necessary)

Question 6

Subjective approach to assigning probabilities

Answer 6

Used when classical approach isn’t reasonable and there is no history of outcomes to study

Probability = degree of belief we hold in the occurrence of an event (potential from the study of related factors

Question 7

Simple event

Answer 7

An individual outcome of a sample space

Question 8

Event

Answer 8

A collection or set of one or more simple events in a sample space

Question 9

Probability of an event

Answer 9

The sum of the probabilities of the simple events that constitute the event

Question 10

Intersection of events A and B

Answer 10

The event that occurs when both events A and B occur. Denoted as: A and B

Probability of intersection = joint probability

Question 11

Joint probability

Answer 11

Probability that a specific intersection of events will occur

Joint probability of all possible combinations = 1

Question 12

Marginal probabilities

Answer 12

Calculated from joint probability tables (in the margins)

Sum of probability of each district event

Question 13

Conditional probability

Answer 13

Probability of one event given the occurrence of another, related, event

P(B1| A1) (probability of B1 given A1)

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

Probability of both events occuring divided by the probability of the given event occuring

Aka ratio of joint probability to marginal probability

Question 14

Independent events

Answer 14

Events are independent if the probability of one event is not affected by the occurances of the other event

If P(A|B) = P(A)

If probability not equal then the events are dependant

Question 15

Union of events

Answer 15

A union of events occurs with either event or both events occur. Denoted as:

A or B

Sum of all probabilities of possible outcomes: A only, B only, A and B

(or 1 minus the probability that neither event occurs)

Question 16

Complement of an event

Answer 16

The event that occurs when event A does not occur (Ac)

Question 17

Complement Rule

Answer 17

Probability of an event and it’s complement must sum to 1

For any event A

Probability of the complement of A = 1 - probability of A

P(Ac) = 1 - P(A)

Question 18

Multiplication rule

Answer 18

Used to calculate the joint probability of two events

P(A and B) = P(B) * P(A|B)
- for dependent events
(Also = P(A) * P(B|A)

For independent events:
P(A and B) = P (B) * P(A)

Question 19

Addition rule

Answer 19

Calculated the probability of the union of two events

(Probability that event A, or event B, or both occur)

P (A or B) = P(A) + P(B) - P (A and B)

(Because the probability of A and B is already included in P(A)+P(B))

Question 20

Addition rule for mutually exclusive evets

Answer 20

P (A and B) = P(A) + P(B)

Question 21

Probability tree

Answer 21

Events in an experiment represented by lines that branch at each choice/instance

Total probability of all ending joint probabilities should be 1

For mutually exclusive events probability trees are added not multipled

Question 22

Without replacement

Answer 22

Options from a limited pool are not replaced after first choice is made, changing the probability of the next pick

(Vs with replacement where probability of a given pick is the same every time because option is replaced)

Question 23

Prior probabilities

Answer 23

Probabilities determined prior to the decision in question (when asking a question if x then what will happen to y these are the probabilities of Y if not X)

Question 24

Likelihood probabilities

Answer 24

Conditional probabilities based on a possible decision

(when asking a question if x then what will happen to y these are the varies probilities of y if decision x is made or not made)

Question 25

Posterior probabilities

Answer 25

Or revised probabilities

(when asking a question if x then what will happen to y, given X or not x)

Revised to reflect answer to decision

Question 26

Bayes law

Answer 26

P(Ai|B) = (P(Ai)P(B|Ai))/ (P(A1)P(B|A1))+P(A2)P(B|A2))+…+(P(An)P(B|An))

Aka
Probability of Ai and B / probability of B