lecture 12

Question 1

describe partition of S with subsets A and B

Answer 1

A ∩ B
A ∩ B^c
A^c ∩ B
A^c ∩ B^c
whole of S into 4 subregions

Question 2

describe what we can deduce from these 4 subregions

Answer 2

(A ∪ B)^c = A^c ∩ B^c
similarily
(A^c ∪ B^c) = (A ∩ B)^c
(A^c ∪ B^c)^c = A ∩ B

Question 3

what are de morgans laws

Answer 3

A^c ∩ B^c = (A ∪ B)^c
(A^c ∪ B^c)=(A ∩ B)^c

Question 4

describe experiment

Answer 4

this can be interpreted as any setting in which an uncertain consequence is to arise;
could involve observing an outcome, taking a measurement etc.

Question 5

describe considering all outcomes of the experiment

Answer 5

Identify all the outcomes that can arise, and denote the corresponding set by S
The set S is termed the sample space of the experiment
The individual elements of S are termed sample points
(or sample outcomes)

Question 6

define an event

Answer 6

An event A is a collection of sample outcomes
That is, A is a subset of S, A is in some or all of S
The individual sample outcomes are termed simple (or
elementary) events, and may be denoted E1,E2,…,Ek…

Question 7

describe terminology

Answer 7

We say that event A occurs if the actual
outcome, s, is an element of A.

Question 8

describe terminology for 2 events = A and B

Answer 8

A ∩ B occurs if and only if A occurs and B occurs =
s is an element of A ∩ B
A ∪ B occurs if A occurs or B occurs or if BOTH A and B occur = s is an element of A ∪ B
if A occurs then A^c does not occur

Question 9

what is the certain event

Answer 9

S
sample outcome resulting from experiment has to be an element of S by definition

Question 10

what is the impossible event

Answer 10

empty set ∅
cannot observe bc empty
none of sample outcomes

Question 11

describe mathematical definition of probability

Answer 11

For event A, P is the function that assigns
P(A)=p
where p is a numerical value
but TOO GENERAL

Question 12

describe 3 rules of probability

Answer 12

S = sample space for experiment, A & B are events (subsets of S)
P(A) observes =

P(A) is greater than or equal to 0 (NON NEGative)
P(S) = 1
IF A ∩ B = ∅ THEN
P(A ∪ B) = P(A) + P(B)

Question 13

informally describe rules of probability

Answer 13

P measures content of A (how likely sample outcome ends up in A)
rules =
content cannot be neg
total content of S standardized to be A
if A and B do not intersect = total contents is sum of individual contents

Question 14

describe the extension of rule 3

Answer 14

If A1, A2, . . . , Ak are a collection of events such that
Aj ∩ Ak = ∅ for all j cannot equal k
then
P (A1 ∪ A2 ∪ … ∪ Ak) = P(A1) + P(A2) + … + P(Ak)
just add all probabilities

Question 15

consequences of rules - for any A P(A^c) =

Answer 15

for any A P(A^c) = 1- P(A)

Question 16

consequences of rules - P(∅) =

Answer 16

P(∅) = 0

Question 17

consequences of rules - for any A P(A) is less than or equal to

Answer 17

for any A P(A) is less than or equal to 1 (must be between 0 and 1)

Question 18

consequences of rules - for any 2 events A and B If A if some or all in B then

Answer 18

for any 2 events A and B If A if some or all in B then
P(A) less than or equal to P(B)

Question 19

consequences of rules - general addition rule

Answer 19

P (A ∪ B)= P(A) + P(B) - P(A ∩ B)
bc double counting intersection so need to subtract it therefore
P (A ∪ B) is less than or equal to P(A) + P(B)

Question 20

describe booles inequality

Answer 20

Probability of union of events when not mutually exclusive
content of union must be less than or equal to sum of individual As
see notes for formula