Basic Properties of Probability
Definitions: A random experiment is a procedure or an operation whose outcome is uncertain
and cannot be predicted with certainty in advance. The collection of all possible outcomes is
called the sample space. We will typically use the letter S to denote a sample space.
An event is any subset of the sample space. Events are usually denoted by capital letters other
than S. Events are collections (or sets) of outcomes from the sample space.
Examples of random experiments:
1. Toss a coin three times and record the results of each toss in order. List all events in the sample
space.
Two examples of events from this random experiment are
A = the event that there are exactly 2 heads in the 3 tosses
B = the event that there are no heads
List the outcomes contained in each of these events.
2. Toss a coin repeatedly and count the number of tosses until the first heads. What is the sample
space?
3. Measure the lifetime of a light bulb in hours. What is the sample space?
An example of an event in this sample space would be the event A that a light bulb lasts at least
500 hours. Write this event using set notation.
The outcomes in a sample space are said to be “equally likely” if they will all occur approximately
equally often in the long run if the random experiment is repeated many, many times. In which of the
examples above does the sample space consist of “equally likely” outcomes?
1
Notation of Set Theory
First let us introduce some definitions and notation:
Definitions: Let S be a sample space and let A and B be any two events from S. Then
 The union of the events A and B, denoted A  B , is the event consisting of all outcomes that
belong to A or B or both.
 The intersection of the events A and B, denoted A  B or sometimes by the shorter AB, is the
event consisting of all outcomes common to both A and B.
 The complement of an event A, denoted Ac , is the collection of all outcomes that are not in A.
 The event A is a subset of B, denoted A  B , if every outcome in A is also contained in B.
 The empty set or null set, denoted Ø, is the event which consists of no outcomes.
 The events A and B are disjoint or mutually exclusive if A and B cannot happen
simultaneously. Thus A and B are disjoint if A  B  .
 Pr(A) or P(A) is used to denote the probability of event A.
Example: Shuffle a standard deck of 52 cards and randomly select one card from the deck. Then the
sample space S consists of each of the 52 cards in the deck. Some possible events to consider:
A = the card is a heart
B = the card is a face card
C = the card is the king of hearts
D = the card is black
Describe the following related events:
Ac =
A B =
A B =
The union of A and B is an event consisting of 22 outcomes (all 13 of the hearts plus the king, queen,
and jack from each of the others suits).
Note that C  A and C  B.
List two pairs of events from above that are mutually exclusive.
Probability as a Long-term Relative Frequency
The probability of a random event is the long run proportion (or relative frequency) of times the event
would occur if the random process were repeated over and over an extremely large number of times under
identical conditions. The probability of an event can be approximated by simulating the process a large
number of times. Simulation leads to an empirical, or experimental, estimate of the probability.
2
Treating probabilities as long-term frequencies is known as the frequentist approach to probability.
Using the frequentist approach, if a sample space consists of a finite number of possible outcomes, say N,
and all outcomes are equally likely, then it is natural to assign equal probabilities to each outcome. That
is,
1
1
P  each outcome  
 .
total number of outcomes N
Furthermore, in this situation in which all outcomes are equally likely, if an event A consists of M distinct
outcomes, then the probability of the event A is given by
P  A 
number of outcomes in A
number of outcomes in S

M
N
.
In the example of dealing cards on the previous page, since we are shuffling the deck and then randomly
selecting one card, all 52 cards are equally likely. As a result, we can compute the probability of any
event in the sample space S simply by counting the number of outcomes in the event and dividing by 52,
the total number of outcomes in the sample space. For example,
3
P(the card is a heart and a face card) = P  A  B   .
52
Find the probability of each of the other events described on the previous page.
Note: Since events are sets, it makes sense to perform set operations such as complement, intersection,
and union on them, but it makes no sense to perform arithmetic operations such as addition or
multiplication on events. On the other hand, probabilities are numbers, so it is legitimate to add,
multiply, and divide probabilities but not to take complements, intersections, or unions of them.
3
Case Study: 100 Best Films
In 1998, the American Film Institute created a list of the top 100 American films ever made. The list is
included below.
Rank Title
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
Citizen Kane
Casablanca
The Godfather
Gone With The Wind
Lawrence Of Arabia
The Wizard Of Oz
The Graduate
On The Waterfront
Schindler's List
Singin' In The Rain
It's A Wonderful Life
Sunset Boulevard
The Bridge On The River Kwai
Some Like It Hot
Star Wars
All About Eve
The African Queen
Psycho
Chinatown
One Flew Over The Cuckoo's Nest
The Grapes Of Wrath
2001: A Space Odyssey
The Maltese Falcon
Raging Bull
E.T The Extra-Terrestrial
Dr. Strangelove
Bonnie And Clyde
Apocalypse Now
Mr. Smith Goes To Washington
The Treasure Of The Sierra Madre
Annie Hall
The Godfather Part Ii
High Noon
To Kill A Mockingbird
It Happened One Night
Midnight Cowboy
The Best Years Of Our Lives
Double Indemnity
Doctor Zhivago
North By Northwest
West Side Story
Rear Window
King Kong
The Birth Of A Nation
A Streetcar Named Desire
A Clockwork Orange
Taxi Driver
Jaws
Snow White And The Seven Dwarfs
Butch Cassidy And The Sundance Kid
Year
1941
1942
1972
1939
1962
1939
1967
1954
1993
1952
1946
1950
1957
1959
1977
1950
1951
1960
1974
1975
1940
1968
1941
1980
1982
1964
1967
1979
1939
1948
1977
1974
1952
1962
1934
1969
1946
1944
1965
1959
1961
1954
1933
1915
1951
1971
1976
1975
1937
1969
Rank Title
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
Year
The Philadelphia Story
From Here To Eternity
Amadeus
All Quiet On The Western Front
The Sound Of Music
M*A*S*H
The Third Man
Fantasia
Rebel Without A Cause
Raiders Of The Lost Ark
Vertigo
Tootsie
Stagecoach
Close Encounters Of The Third Kind
The Silence Of The Lambs
Network
The Manchurian Candidate
An American In Paris
Shane
The French Connection
Forrest Gump
Ben-Hur
Wuthering Heights
The Gold Rush
Dances With Wolves
City Lights
American Graffiti
Rocky
The Deer Hunter
The Wild Bunch
Modern Times
Giant
Platoon
Fargo
Duck Soup
Mutiny On The Bounty
Frankenstein
Easy Rider
Patton
The Jazz Singer
My Fair Lady
A Place In The Sun
The Apartment
Goodfellas
Pulp Fiction
The Searchers
Bringing Up Baby
Unforgiven
Guess Who's Coming To Dinner
Yankee Doodle Dandy
1940
1953
1984
1930
1965
1970
1949
1940
1955
1981
1958
1982
1939
1977
1991
1976
1962
1951
1953
1971
1994
1959
1939
1925
1990
1931
1973
1976
1978
1969
1936
1956
1986
1996
1933
1935
1931
1969
1970
1927
1964
1951
1960
1990
1994
1956
1938
1992
1967
1942
Suppose that two people (we’ll call them Allan and Beth) get together to watch a movie and, to avoid
potentially endless debates about a selection, decide to choose a movie at random from the “top 100” list.
You will investigate the probability that it has already been seen by at least one of them.
4
Explorations
Basic Probability Rules
Suppose that one film is selected at random from the list. Let A denote the event that Allan has seen the
film and let B denote the event that Beth has seen the film. Note that the events A and B can be thought of
as sets; for example, A is the set of all films that Allan has seen.
Since the movie is being selected “at random,” each of the 100 films is equally likely to be chosen; that is,
each has probability 1/100. The probabilities of various events can thus be calculated by counting how
many of the 100 films comprise the event of interest. For example, the following 2x2 table classifies each
movie according to whether it was seen by Allan and whether it was seen by Beth:
Beth yes
36
16
Allan yes
Allan no
Total
Beth no
9
39
Total
100
a) Translate the following events into set notation using the symbols (A and B, complement, union,
intersection) defined on the previous page. Then give the probability of the event as determined
from the table:

Allan and Beth have both seen the film.

Allan has seen the film and Beth has not.

Beth has seen the film and Allan has not.

Neither Allan nor Beth has seen the film.
b) Fill in the marginal totals of the table. From these totals determine the probability that Allan has
seen the film and also the probability that Beth has seen the film. (Remember that the film is
chosen at random, so all 100 are equally likely.) Record these probabilities along with the
appropriate symbols below.
c) Determine the probability that Allan has not seen the film. Do the same for Beth. Record these,
along with the appropriate symbols, below.
5
d) If you had not been given the table, but instead had merely been told that P  A   .45 and
  and P  B  ?
P  B   .52 , would you have been able to calculate P Ac
c
Explain how.
One of the most basic probability rules is the complement rule, which asserts that the probability of
the complement of an event equals one minus the probability of the event:
 
P Ac  1  P  A  .
e) Add the counts in the appropriate cells of the table to calculate the probability that either Allan or
Beth (or both) have seen the movie. Also indicate the symbols used to represent this event.
f) If you had not been given the table but instead had merely been told that P  A   .45 and
P  B   .52 , would you have been able to calculate P  A  B  ? Explain.
g) One might naively think that P  A  B   P  A   P  B  . Calculate this sum, and indicate whether
it is larger or smaller than P  A  B  and by how much. Explain why this makes sense, and
indicate how to adjust the right side of this expression to make the equality valid.
6
The addition rule asserts that the probability of the union of two events can be calculated by adding
the individual event probabilities and then subtracting the probability of their intersection:
P  A  B   P  A  P  B   P  A  B  .
This rule should make good intuitive sense. If we simply add the probabilities of events A and B, we are
counting all outcomes that are in both A and B twice, so we need to subtract off the probability of A  B in
order to eliminate this double counting.
h) Use this addition rule as a second way to calculate the probability that Allan or Beth has seen the
movie, verifying your answer to e).
i)
As a third way to calculate this probability, first identify (in words and in symbols) the complement
of the event that Allan or Beth has seen the movie. Then find the probability of this complement
from the table. Then use the complement rule to determine P  A  B  . Does this match your
answers to e) and h)?
j)
Under what circumstances is it valid to say that P  A  B   P  A   P  B  ?
If A  B   , then it follows that P  A  B   P  A   P  B  . This is known as the addition rule for
disjoint events; it is a special case of the addition rule since if A  B   , P  A  B   P     0 .
7
Conditional Probability
We are often interested in the conditional probability of one event given the information that another
event has occurred. You will find that it is straightforward and intuitive to derive a reasonable definition of
conditional probability by examining data in a two-way table.
Suppose again that one of the top 100 films is to be selected at random, and consider again the 2x2 table
indicating how many of the top 100 films Allan and Beth have seen:
Allan yes
Allan no
Total
Beth yes
36
16
52
Beth no
9
39
48
Total
45
55
100
k) Recall the (unconditional) probability that Allan has seen the movie and the (unconditional)
probability that Beth has seen it. Also recall the probability that they both have seen the film.
Record these, along with the appropriate symbols, below:
l)
Now suppose that once the film has been selected, you learn the partial information that Allan has
seen it. Given this information, determine the conditional probability that Beth has seen it by
restricting your attention to the “Allan yes” row of the table and assuming that those films are
equally likely.
We use the notation P  B A  to denote the conditional probability of event B given event A.
m) Determine how P  B A  relates to P  A  , P  B  , and P  A  B  . [Hints: Actually, one of these
three probabilities is irrelevant. Determine which two are relevant and how they relate to P  B A 
by following your calculation from the table in (l).
8
Definition: The conditional probability of event B given event A is defined as follows:
P  A  B
,
P  B A 
P  A
provided that P  A   0 .
Note that when defining a conditional probability, it is essential to require that P  A  is positive since it
would not make sense to condition on an event that is impossible in the first place.
n) Use the definition to calculate the conditional probability that Allan has seen the movie given that
Beth has seen it. Does the knowledge that Beth has seen it increase, decrease, or not affect the
(unconditional) probability that Allan has seen it? Explain.
o) Does P  B A   P  A B  in this case? To convince yourself that these need not even be close,
consider selecting one American citizen at random. Let M be the event that the person is male, and
let S be the event that the person is a U.S. Senator. Make an educated guess as to the values of
P  M S  and P  S M  . Are they close?

p) Use the definition of conditional probability to calculate P B Ac
 and P  A B  for the film
c
example. Does the knowledge that Beth has not seen the film increase, decrease, or not affect the
probability that Allan has seen it? Explain.
Independence
Two events are said to be independent if knowledge that one occurs does not change the probability of
other’s occurrence. In other words, the events are independent if the conditional probability of one given
the other (e.g., P  A B  ) is the same as the unconditional probability of the one in the first place (e.g.,
P  A  ). Thus, in symbols, events A and B are independent if P  A B   P  A  . This is equivalent to
requiring that P  B A   P  B  .
9
q) Express this condition for independence in terms of the probability of the intersection of A and B.
[Hint: Use the definition of conditional probability on either of the expressions above.]
You should find that another equivalent definition for A and B to be independent is that
P  A  B   P  A   P  B  . Mathematicians typically take this as the definition of independence and then
prove that this is equivalent to the other two conditions for probability given above.
Definition: Two events A and B are independent if and only if P  A  B   P  A  P  B  .
The following theorem follows from the definition above and basic properties of conditional probabilities:
Theorem: Let A and B be any two events. Then the following are equivalent: (That is, if any one of the
following statements is true, then all four must be true.)
1. A and B are independent,
2. P  A  B   P  A  P  B  ,
3. P  A B   P  A  ,
4. P  B A   P  B  .
As a consequence of this theorem, we only need to check any one of the conditions above to check for
independence of two events.
r) Are the events {Allan has seen the film} and {Beth has seen the film} independent? Defend your
answer using any one of the equivalent definitions of independence. Then write a sentence or two
explaining why your answer makes sense given the data in the table.
Example: Randomly select one card from a standard deck of 52. Consider the following three events:
A = the card is a heart
B = the card is a face card
C = the card is the king of hearts
13 1
 . Similarly, there are 12 face cards (4 jacks, 4
First note that there are 13 hearts, so P  A  
52 4
12 3
1
queens, 4 kings), so P  B  
 . Finally, there is only one king of hearts, so P  C   .
52 13
52
10
If we know that the event A has occurred, then we know that the card is one of the 13 hearts. With
3
this knowledge that the card is a heart, the probability of the event B becomes P  B | A   . Note
13
3
 P  B  . Thus A
that this is the same as the unconditional probability of B; that is, P  B | A  
13
and B are independent – knowing that the card is a heart has no effect at all on the probability that
the card is a face card. Similarly, knowing that the card is a face card has no effect at all on the
probability that it is a heart (that is, P  A | B   P  A  ; verify this for yourself.)
On the other hand, if we know that the event B has occurred, then we know that the card is one of
1
the 12 face cards, so P  C | B   . This is not the same as the unconditional probability P  C  ,
12
so B and C are not independent (they are dependent). In this case, knowing that the card is a face
card substantially increases the probability that it is the king of hearts – knowing that on of the two
events has occurred does have a strong impact on the probability of the other occurring – so the are
not independent. Finally, note that if we know that the event C has occurred, then the card is a face
card, so P  B | C   1 and P  B | C   P  B  , so we have verified in a second way that B and C are
not independent.
Axioms of Probability and Proofs of Basic Probability Rules
In the explorations section above, we used intuition to discover rules for probability that seem to make
sense. In fact, each of these rules can be proven using the axiomatic approach to probability. In the
axiomatic approach, developed by the Russian mathematician Kolmogorov (1903-1987), we begin with a
small number of axioms (or assumptions). These axioms are assumed to be self-evident, and then, on the
basis of a few definite rules of mathematical and logical manipulation, all other results are carefully proven
or derived from these axioms.
Probability theory is based on the following three axioms. Let S denote the sample space of an experiment.
Associated with each event A in S is a number P(A), called the probability of A which satisfies the
following axioms:
Axiom 1: P  A   0 for every event A.
Axiom 2: P  S   1.
Axiom 3: If A and B are mutually exclusive events, then P  A  B   P  A   P  B  .
Axiom 3a: (Needed if sample space is infinite.) If A1 , A2 , A3 ,  is an infinite sequence of mutually

 j 1



exclusive events, then P   Aj    P  Aj  .
j 1
These axioms can be used to prove many of the other results discovered earlier in this chapter.
 
Theorem 1: (Complement Rule) P Ac  1  P  A  .
Proof:
11
Theorem 2: P     0.
Proof: Let A = S. Then Ac   . Now apply Theorem 2.1 and Axiom 2 to obtain
P     P  Ac   1  P  A   1  P  S   1  1  0.
Theorem 1
Axiom 2
Theorem 3: If A and B are events and A  B, then P  A   P  B  .
Proof:
Theorem 4: For every event A, 0  P  A   1.
Proof:
Theorem 5: (Addition Rule for Two Events) For any two events A and B,
P  A  B   P  A  P  B   P  A  B  .
Proof:
12
Corollary: (Bonferroni Inequality) For any two events A and B,
P  A  B   P  A  P  B  .
Proof: By Axiom 1, P  A  B   0, so  P  A  B   0. Using this inequality along with the result
of Theorem 2.5, we have
P  A  B   P  A   P  B   P  A  B   P  A   P  B   0. □
Theorem 6: (Addition Rule for Three Events) For any three events A, B, and C,
P  A  B  C   P  A  P  B   P  C   P  A  B   P  A  C   P  B  C   P  A  B  C  .
Proof: To prove this, write A  B  C  A   B  C  and then use Theorem 2.5 twice. The
details are left as an exercise.
Theorem 7: Let A and B be any two events. Then the following are equivalent:
1. A and B are independent,
2. P  A  B   P  A  P  B  ,
3. P  A B   P  A  ,
4. P  B A   P  B  .
Proof: Numbers (1) and (2) are equivalent by the definition of independence.
We will prove here that (2) and (3) are also equivalent. The proof that (2) and (4) are
equivalent is left as an exercise. Since all of the other statements are equivalent to (2), all
four are equivalent.
We begin by showing that (2) implies (3). By the definition of conditional probability,
P  A  B
P  A | B 
. Assuming that (2) is true, this can be rewritten as
P  B
P  A P  B 
 P  A  . Thus if (2) is true, then (3) must also be true.
P  B
To prove that (2) and (3) are equivalent, we also need to prove the converse of the
statement above. That is, we need to prove that if (3) is true, then (2) must also be true. By
P  A  B
the definition of conditional probability, P  A | B  
. Assuming that (3) is true,
P  B
P  A | B 
we can rewrite this as P  A  
P  A  B
, so multiplying by P  B  yields the desired
P  B
result: P  A  B   P  A  P  B  . Thus (3) implies (2), completing the proof. □
Theorem 8: If A and B are independent events, then each of the following pairs of events are also
independent:
1. A and B c
2. Ac and B
3. Ac and B c
13
Proof: We will prove (1) here. The proofs of (2) and (3) are left as an exercise.
Assume that A and B are independent. Note that the event A can
be broken into two parts: the part of A that is inside of B, which is
A
B
denoted A  B or just AB, and the part of A that is outside of B,
c
which is denoted A  B c or just ABc.
AB AB
So A  AB  AB c . Furthermore, the events AB and ABc are clearly
mutually exclusive (an outcome cannot be both in B and not in B).
Now, using Axiom 3 along with the fact that AB and ABc are
mutually exclusive, we obtain
P  A   P  AB  AB c   P  AB   P  AB c  .
Since A and B are independent, P  AB   P  A  P  B  , so the equation above becomes
P  A   P  A P  B   P  AB c  .
Subtracting P  A  P  B  from both sides and then factoring yields
P  AB c   P  A   P  A  P  B   P  A  1  P  B   .
Finally, by the complement rule (Theorem 2.1), 1  P  B   P  B c  , so we have
P  AB c   P  A  P  B c  .
But this means that A and B c are independent (by the definition of independence). □
Exercises
1. Suppose that you flip two fair coins. Is the sample space of equally likely outcomes properly
represented as {HH, TH, HT, TT} or as {2 heads, 2 tails, 1 of each}? Explain.
2. For each of the following situations, list the sample space (that is, list all possible outcomes) and also
indicate whether it seems reasonable to assume that all of the outcomes are equally likely. If not,
include a short explanation.
a) whether or not you pass this course
b) your grade in this course
c) the color of a randomly selected M&M candy
d) the outcome of the roll of a fair die
e) the sum of the outcomes of independently rolling two fair dice
f) a tennis racquet landing with the label “up” or “down” when spun
g) the last digit of the Social Security Number of a randomly selected American
h) the number of flips of a fair coin until the first “heads” appears
3. Suppose that you independently roll two fair four-sided dice. (Each die is equally likely to land on 1
or 2 or 3 or 4.)
a) Using the notation (x,y) to mean that the first die lands on x and the second on y, list all 16
outcomes in the sample space.
b) List all of the outcomes in the following events:
o A = {the first die lands on 2}
o B = {the sum of the dice exceeds 5}
o C = {the first die lands on a larger number than the second die}
o D = {the difference between the two dice is one or less}
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c) Determine the probability of each event listed in b).
d) Now suppose that the two dice were fair but six-sided. Recalculate the probabilities of the
events listed in b). (You need not list out all of the outcomes in each event.)
4. Identify which of the following are legitimate uses of event/probability notation and which are not.
Give an explanation for the ones that are not.
d) P  A  B   P  A  B 
a) P  A  B 
b) P  A   P  B 
c)

P A  Bc

 A  B  
c
e)
P
f)
 P  AB  
c
5. Suppose that you roll two fair, six-sided dice. Consider the events:





A = {the first die lands on 2}
B = {the sum of the dice equals 7}
C = {the first die lands on a larger number than the second die}
D = {the difference between the two dice is one or less}
E = {the sum of the dice equals 11}.
a) Identify all pairs of these events that are disjoint. Explain your answers.
b) Identify all pairs of these events that are independent. Justify your answers with appropriate
calculations.
c) Identify one pair of these events with the property that learning that one has occurred makes the
other more likely to have occurred. Justify your answer with appropriate calculations.
d) Identify one pair of these events with the property that learning that one has occurred makes the
other less likely to have occurred. Justify your answer with appropriate calculations.
6. Suppose that you hear a weather forecast announcing that the probability of rain on Saturday is
50% and that the probability of rain on Sunday is 50%. Define the events A = {rain on Saturday}
and U = {rain on Sunday}.
a) If A and U are independent, what is the probability of rain on at least one day of the
weekend?


b) If P U A   .8 , what must be true of P U Ac ? In this case what is the probability of rain
on at least one day of the weekend?
c) What is the largest possible value for P  A  U  ? What has to be true of P U A  to achieve
this value?
d) What is the smallest possible value for P  A  U  ? What has to be true of P U A  to
achieve this value?
7.
Given the following probabilities:
P  A   0.5
P  B   0.6
P  A  B   0.3
P  A  B  C   0.1
P  C   0.6
P  A  C   0.2
P  B  C   0.3




find the conditional probabilities P  A B  , P  B C  , P A B c , and P B c A  C .
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8.
Let A and B be two events and let P  A   0.4 , P  B   p , and P  A  B   0.8 .
a. For what value of p will A and B be mutually exclusive?
b. For what value of p will A and B be independent?
9.
A survey is conducted to determine the sources that people in a large metropolitan area use to get
news. The survey indicates that 77% obtain news from television, 63% from newspapers, 47%
from radio, 45% from television and newspapers, 29% from television and radio, 21% from
newspapers and radio, and 6% from all three.
a. Sketch a Venn diagram and fill in all of the appropriate probabilities.
b. What proportion of people obtain news from television, but not newspapers?
c. What proportion of people do not obtain news from either television or radio?
d. What proportion of people do not obtain news from any of these three sources?
e. Given that radio is a news source, what is the probability that a newspaper is also a news
source?
f. Given that TV is a news source, what is the probability that radio is not a news source?
g. Given that both newspaper and radio are news sources, what is the probability that TV is
not a news source?
10. Prove Theorem 2.6. Make sure that every step of your proof is clearly explained and justified.
Indicate all other theorems or axioms that you make use of in your proof.
11. Prove parts (2) and (3) of Theorem 2.8. Make sure that every step of your proof is clearly
explained and justified. Indicate all other theorems or axioms that you make use of in your proof.
12. Show that the 3 axioms of probability are satisfied by conditional probabilities. In other words, if
P  B   0, prove that
a.
P  A | B   0,
b.
P  B | B   1,
c. If A1 and A2 are mutually exclusive, then P  A1  A2 | B   P  A1 | B   P  A2 | B  .
Make sure that every step of your proof is clearly explained and justified. Indicate all theorems or
axioms that you make use of in your proof.
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