Chapter 2: Probability

Question 1

Experiment

Answer 1

Any activity or process whose outcome is uncertain

Question 2

Sample Space (S)

Answer 2

• The collection of all possible outcomes of a probability experiment
• S​ = {Tails, Heads}

Question 3

Event

Answer 3

• Any collection of outcomes from S
  
  • S itself is an event, the largest one that can be formed
  • Simple Event - consists of a single outcome (tossing a single die)
  • Compound Event - consists of more than 1 outcome (tossing 2 dice)

Question 4

AUB

Answer 4

• (A union B) read as “A or B”
• is the event consisting of all outcomes that are either in A or in B or in both events
  
  • i.e. all outcomes in at least one of the events

Question 5

A intersection B

Answer 5

• read as “A and B”
  
  • is the event consisting of all outcomes are in both A and B

Question 6

A Complement

Answer 6

• A^’ or A^c read as “A prime” or “A complement”
• set of all outcomes in sample space S that are not contained in A

Question 7

Null Event

Answer 7

• ø read as “phi” denotes the null event (event consisting of no outcomes)
  
  • if A intersection B = ø, then A and B are said to be mutually exclusive or disjoint events

Question 8

Mutually Exclusive

Answer 8

• Two events that are mutually exclusive (also called disjoint) if they do not have any elements in common OR it is impossible for them to occur together
• Formally, A and B are mutually exclusive events if and only if (A intersection B)= ø, where ø represents the null event.

Question 9

Complementary Events

Answer 9

• If the union of two mutually exclusive events is the sample space S, they are called complementary events

Question 10

Independent Events

Answer 10

• Two events are independent if the occurrence of one of the events gives us no information about whether or not the other event will occur
  
  • i.e. the events have no influence on each other
• If two events are independent then they cannot be mutually exclusive (disjoint), and vice versa

Question 11

Axioms

Answer 11

1. For any event A, P(A) >= 0
2. P(S) = 1
3. If A₁, A₂, A₃, …, is an infinite collection of disjoint events , then P(A₁ U A₂ U A₃ U …) = Σ​P(A_i)

• P(ø) = 0

Question 12

Objective Interpretation of Probability

Answer 12

• The objective interpretation of probability identifies this limiting relative frequency with P(A)

Question 13

Limiting (long-run) Relative Frequency

Answer 13

• As n gets arbitrarily large, n(A)/n approaches a limiting value

Question 14

Relative Frequency

Answer 14

• The ratio n(A)/n is called the relative frequency of occurrence that event A in the sequence of n replications
• As the number of replications increases, the relative frequency of the event begins to stabilize

Question 15

Subjective Interpretation of Probability

Answer 15

• Interpretations of probability in such situations, where an event is unrepeatable, are thus referred to as subjective interpretations of probability
• e.g. “The chances are good for a peace agreement”
• e.g. “It is likely that our company will be awarded the contract”

Question 16

More Probability Properties

Answer 16

• For any event A, P(A) <= 1
• When events A and B are mutually exclusive events,
  
  • P(A U B) = P(A) + P(B)
• When events A and B are not mutually exclusive events,
  
  • P(A U B) = P(A) + P(B) - P(A ∩ B)
• For any three events A, B, and C
  
  • P(A U B)=P(A) + P(B) + P(C) - P(A ∩ B) - P(A ∩ C) - P(B ∩ C) + P(A ∩ B ∩ C)

Question 17

Equally Likely Outcomes

Answer 17

Question 18

The Product Rule for Ordered Pairs

Answer 18

• By an ordered pair, we mean that, if O₁ and O₂ are objects, then the pair (O₁ and O₂) is different from the pair (O₂, O₁)
• e.g. (American, United) and (United, American)
• If the first element or object of an ordered pair can be selected in n₁ ways, and for each of these n₁ ways, the second element of the pair can be selected in n₂ ways, then the number of pairs is n₁n₂.

Question 19

Product Rule for k-Tuples

Answer 19

• •We will call an ordered collection of k objects a k-tuple
  (so a pair is a 2-tuple and a triple is a 3-tuple).
• Total number of 3-Tuples is N=n₁n₂n₃ = (5)(12)(9) = 540 ways to choose first an appliance dealer, then a plumbing contractor, and finally an electrical contractor

Question 20

Permutation

Answer 20

• An ordered subset of size k, from a total of n objects
  
  • the number of permutations of size k that can be formed from the n individuals or objects in a group will be denoted P _k,n
• P_3,7 = (7)(6)(5) = 210
• Total permutations of size 3 out of 7 items = 3! x total number of combinations

Question 21

Combination

Answer 21

• An unordered subset of size k, from a total of n objects
• Denoted by C_k,n

1. C_0,n = 1
2. C_1,n = n
3. C_n,n = 1

Question 22

Relationship between Permutations and Combinations

Answer 22

Question 23

Conditional Probability

Answer 23

• P(A|B) = conditional probability of A given that the event B has occurred
• P(A) is the original, or unconditional probability, of event A
• For any two events A and B with P(B) > 0, the conditional probability of A given that B has occurred is defined by

Question 24

The Multiplication Rule for P(A ∩ B)

Answer 24

• This rule is important because it is often the case that
  P(A ∩ B) is desired, whereas both P(B) and P(A|B) can be specified from the problem description.

Question 25

Bayes’ Theorem

Answer 25

Question 26

The Law of Total Probability

Answer 26

• Let A₁, …, A_k be mutually exclusive and exhaustive events

Question 27

Independence

Answer 27

• Two events A and B are independent if P(A|B) = P(A) and are dependent otherwise
• Multiplication Rule for P(A∩B)
  
  • A and B are independent if and only if:
    
    • P(A∩B) = P(A|B) • P(B) = P(A) • P(B)