Probability

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Probability
The word probability derives from the Latin
probare (to prove, or to test). Informally,
probable is one of several words applied to
uncertain events or knowledge, being closely
related in meaning to likely, risky, hazardous,
and doubtful. The theory of probability is the
branch of mathematics that studies chances and
the long-term patterns of random outcomes.
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How do we interpret 70 chance of precipitation?
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Have you ever wondered how gambling be business
for the casino? It is a remarkable fact that the
aggregate result of many thousands of random
outcomes can be known with near
certainty. Individual gamblers can never say
whether a day at the casino will turn a profit or
a loss. But the casino is not gambling. It does
not need to load the dice, mark the cards, or
alter the roulette wheel. It knows that in the
long run each dollar bet will yield it five cents
or so of revenue.
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The scientific study of probability is a modern
development. It began with the study of chances
in games and gambling. The cofounders of
probability theory, Pierre de Fermat and Blaise
Pascal (1654) were proposed of the Gamblers
Dispute problem by a gambler, Chevalier de Méré,
who wanted to know whether the payoff in a
certain game is fair (more on this later).
Nowadays probability plays the key role in casino
business, but it is also heavily used in
insurance business, economics, and industry.
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In our classroom however, we still use coins,
cards, dice, and wheels as examples because their
mathematical models are easy to define and their
associated experiments can be performed hundreds
of times in the classroom. On the other hand,
the mathematical model for life insurance
companies is very complicated and we cannot
perform experiments in the classroom to collect
data for checking the correctness of that model.
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Relative Frequency
If we flipped a normal coin 1000 times and
observed that it landed on head 465 times, then
we say that the relative frequency of getting a
head is 465/1000 0.465 in these 1000
repetitions of the experiment. In another terms,
we can also say that the empirical probability
of getting a head is 0.465 (in these 1000
repetitions of the experiment).
The original purpose of building a mathematical
theory of probability is to find a formula that
can predict the empirical probability of any
event to a high accuracy. In addition, we want
the prediction to be more accurate when the
number of repetitions increases. If we
succeed, we can save many hours from repeating
the same experiment millions of times.
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Note Relative frequency Empirical probability
Experimental probability
These are just different terms for the same thing.
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Mathematical models for Probability
For any specific experiment (or random
phenomenon) E, it sample space S is the set of
all possible outcomes in that experiment.
Example 1. Flipping a coin In this case, if we
do not allow the coin to land on its edge, there
will be only two outcomes. Hence the sample space
S Head, Tail (Note that the size of the
sample space is 2, but the sample space itself is
not 2.)
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Mathematical models for Probability
For any specific experiment (or random
phenomenon) E, it sample space S is the set of
all possible outcomes in that experiment.
Example 2. Rolling a (6-face) die If we observe
just the face landing on top there will be 6
possible outcomes. Hence the sample space
S 1, 2, 3, 4, 5, 6 (Again, note that
the size of the sample space is 6, but the sample
space itself is not 6.)
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Mathematical models for Probability
For any specific experiment (or random
phenomenon) E, it sample space S is the set of
all possible outcomes in that experiment.
Example 3. Spinning a Roulette wheel There are
38 slots for the ball to drop into. Hence the
sample space S 00, 0, 1, 2, 3, 4,
5, 6, , 35, 36
(Again, note that the size of the sample space is
38, but the sample space itself is not 38.)
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Mathematical models for Probability
For any specific experiment (or random
phenomenon) E, it sample space S is the set of
all possible outcomes in that experiment.
Example 4 2 cards are drawn simultaneously from
the following set.
The sample space will then be S AK, AQ,
AJ, A10, A9, A8, , KQ, KJ, K10,
,
, 43, 42, 32 and there
should be 78 elements in S.
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Mathematical models for Probability
A probability model for an experiment E is a
mathematical description of E consisting of two
parts a sample space S and a way of assigning
probability to its outcomes.

• Rules
• The probability of any outcome is a number
  between 0 and 1.(and the probability of an
  impossible outcome must be 0.)
• All possible outcomes together must have
  probability 1.

Example 1. Flipping a coin. If the coin
is fair and the person is flipping the coin
randomly, then we believe that the head is equal
likely to land on top as the tail. Hence
p(Head) p(Tail)
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Mathematical models for Probability
A probability model for an experiment E is a
mathematical description of E consisting of two
parts a sample space S and a way of assigning
probability to its outcomes.

• Rules
• The probability of any outcome is a number
  between 0 and 1.(and the probability of an
  impossible outcome must be 0.)
• All possible outcomes together must have
  probability 1.

Example 2. Rolling a die. If the die is
fair, then each is equal likely to land on top.
Hence p(1 on top) p(2 on
top) p(6 on top)
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Casino Dice are carefully machined, and their
drilled holes, called pips, are filled with white
material in density equal to the plastic body.
This guarantees that the side with 6 pips has the
same weight as the opposite side which has only
one pip.
Thus each side is equally likely to land upward.
All the odds and playoffs of dice games depends
on this carefully planned randomness.
Dice balancing Caliper.
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Mathematical models for Probability
An event A is any single outcome or a collection
of outcomes in the experiment. In other words, it
is a subset of the sample space S. The
probability of an event A, p(A), is the sum of
the probabilities of all the outcomes in A.
Example 1 Let us roll a fair 6-face die, and let
A be the event of getting an even number on top.
Then p(A)
In particular, if every outcome in the experiment
is equal likely to occur (which is very common
assumption), then p(A)
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Mathematical models for Probability
Example 2 Let us drop a ball in to a turning
roulette, and let A be the event of getting a
number between 1 and 18 inclusive.
Since the roulette is almost perfectly balanced,
every outcome in the experiment is equal likely
to occur p(ball landing
on any specific number) Hence p(between 1
and 18)
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Mathematical models for Probability
Example 3. Spinning the pointer of the following
wheel. The sample space S red, yellow,
green, blue
If the pointer is perfectly balanced and the
bearing is very smooth, then it is equally likely
to stop at any position, hence p(red)
p(blue)
p(green)
p(yellow)
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0.3
0.6
0
0.4
1
0.7
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The law of Averages (also called the Law of large
numbers) Consider an experiment E in which
the theoretical probability of an event A is p.
Suppose that the single trial of this experiment
is repeated many times, and that the outcome of
each trial is independent of the others. If the
number of trials increases, the experimental
probability of A will approach the theoretical
value p.
Example Suppose that the theoretical prob of
winning a game X is 26, then
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A famous puzzle
In America during the gold-rush era, a very
ingenious gambling game garnered a lot of money
for its perpetrators. Three cards were placed in
a hat one was gold on both sides, one was
silver on both sides, and one was gold on one
side and silver on the other side. The gambler
would take one card and place it on the table
showing (for instance) gold on the top side of
the card. Then he would bet the on lookers even
money that gold would be on the reverse side, his
reasoning being that the card (on the table)
could not be sliver/silver, hence there were only
two possibilities gold/silver or gold/gold. A
fair and even bet, isnt it?
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State Lotteries
The most popular game in state lotteries is
Lotto. By 2006, there are only 7 states without
Lottos.
To play the California Super Lotto plus you need
to pick 5 number from 1 to 47 and one mega number
from 1 to 27. Prior to June 6, 2000, the format
was to pick 6 numbers from 1 to 49, hence the
nick name 649. This format is still being used in
many states.
Lottos are a bad bet, because the state pays out
only about half of the money wagered. The only
compensation almost all Lotto players receive is
the pleasure of dreaming themselves rich.
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Raffle vs. Lottery

• there may not be a winner
• the prob of winning is fixed
• several tickets can share the same grand
  prize.
• the chance of winning is usually extremely
  small.

• there must be a winner
• the prob of winning depends on the number of
  tickets sold
• only one winner per prize

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Area Models for Probability
When a student was walking across the room (with
tiles on the floor as illustrated below), a small
button fell off from her dress. What is the
probability that the (center of the) button
landed on a blue tile?
Answer Since there are totally 80 square tiles
and 20 of them are blue, hence the probability of
landing on a blue tile should be
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The following target is made up of concentric
circle with radii 1, 2, 3, and 4 units. If a dart
was thrown randomly and hit the target, what is
the probability that it hit the red ring?
Answer area of target p(4)2 16 p area of
red ring p(3)2 p(2)2
5p
4
3
2
Hence Prob(hitting red ring)
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Tree Diagrams
In some experiments it is inefficient to
list all the outcomes in the sample space.
Therefore, we develop alternative procedures to
compute probabilities such as drawing a tree
diagram.
A Tree diagram is a diagram consisting of
line segments connected like the branches and
twigs of a tree. In particular, there is never a
loop in a tree diagram. The starting point of
a tree diagram is called the root. Each
branching point is called a node.The number of
levels in a tree diagram is equal to the number
of steps in the corresponding experiment.
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3rd time
2nd time
1st time
Tree Diagram for the experiment a
coin is flipped 3 times
start
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A coin is flipped 5 times
H
start
T
The orange path represents the sequence of THTHH
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Probability trees and one stage experiments If
we label each branch of the tree with the
appropriate probability, then we get a
probability tree.
red
A ball is drawn from the following jar at random.
2/9
3/9
start
green
4/9
blue
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Complex Experiments and Probability Trees
Start
A Jar contains 2 red balls,
3 green balls, and 4 blue balls. If
two balls are taken out sequentially and randomly
without replacement, what is the probability of
getting two balls of the same color?
Blue
Red
Green
Reset
Reset
Reset
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The Los Angeles Lakers and Portland Trailblazers
are going to play a best 2 out of 3 series.
Suppose that the probability that the Lakers win
an individual game with Portland is 3/5, draw a
probability tree to show possible outcomes.
LA
3/5
LA
3/5
LA
3/5
2/5
P
P
2/5
3/5
LA
LA
3/5
2/5
P
P
2/5
2/5
P
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Multiplicative Property of Probability
Suppose that an experiment consists of a
sequence of simpler experiments. Then the
probability of each final outcome is equal to the
product of the probabilities of the simpler
experiments that make up the sequence.
Example Suppose that we roll a regular die
twice. Then Prob(rolling a 3 followed by
rolling a 5)
Prob(rolling a 3) Prob(rolling a 5)

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Additive Property of Probability
Suppose that an event A is the union of two (or
more) mutually exclusive simpler events A1, A2.
Then Prob(A) Prob(A1) Prob(A2)
Example Suppose that we roll 2 dice
simultaneously. Then Prob(getting a sum of
11) Prob(rolling a 5 on the 1st die and
rolling a 6 on the 2nd die) Prob(rolling
a 6 on the 1st die and rolling a 5 on the 2nd
die)
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Independent Events

• Two events A and B are independent events if the
  occurrence of either event will in no way affect
  the probability of occurrence of the other.
• Examples
• Event A is rolling a sum of 7 from a pair of
  dice, and event B is flipping a head in a
  coin.
• Event A is winning the super lotto, event B is
  winning in a horse race in Del Mar.

Multiplication rule If events A and B are
independent, then the probability that both
events occur (either simultaneously or one after
the other) is
P(A and B) P(A)P(B)
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"A gambler's dispute in 1654 led to the creation
of a mathematical theory of probability by two
famous French mathematicians, Blaise Pascal and
Pierre de Fermat. Antoine Gombaud, Chevalier de
Méré, a French nobleman with an interest in
gaming and gambling questions, called Pascal's
attention to an apparent contradiction concerning
a popular dice game. The game consisted in
throwing a pair of dice 24 times the problem was
to decide whether or not to bet even money (i.e.
1 to 1 payoff) on the occurrence of at least one
"double six" during the 24 throws.
A seemingly well-established gambling rule led de
Méré to believe that betting on a double six in
24 throws would be profitable, but his own
calculations indicated just the opposite.
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Section 11.3 Additional Counting Techniques
Permutations An ordered arrangement of objects is
called a permutation. For example, the
permutations of the letters C,A,S,T are
ACST CAST SACT TACS ACTS CATS SATC TASC ASCT
CSAT SCAT TCAS ASTC CSTA SCTA TCSA ATCS CTAS STA
C TSAC ATSC CTSA STCA TSCA You can see that
there are 4321 24 many permuations.
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Section 11.3 Additional Counting Techniques
Fundamental Counting Property If an event A can
occur in r ways, and for each of these r ways, an
event B can occur in s ways, then event A and B
can occur, in succession, in rs ways.
this condition can also be rephrased as the
number of outcomes in event B is independent
of event A.
Example Suppose that in a local diner, a supper
consists of a starter, an entrée, and a beverage.
If there are 3 choices for the starter, 5
choices for the entrée, and 7 choices for
beverages, how many different suppers can be
created?
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Examples 1. The license plates in Utah consist of
3 digits followed by 3 letters. How many
such license plates are possible?
Passenger vehicle License Plates in California
If the 1st and the 3rd letters cannot be an I and
O, how many possible combinations are there?
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2. Given the set of digits 5, 6, 7, 8, 9, how
many 4-digit numbers can be formed such
that a) the digits are different? b) the
digits are different and the number is divisible
by 5? c) the digits are different and the
number is gt 6000? d) the digits are different
and the number is lt 8000?
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Theorem The number of permutations for n
different objects is
1234 n
The factorial notation The product 1234
n is called n factorial and is written as n!.
Examples 1! 1 2! 12 2 3! 123
6 10! 3,628,800
Remark 0! is defined to be 1.
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Examples 1. Miss Murphy wants to seat her 12
students in a row for a class photo. How
many different seating arrangements are there?
Answer 12!
2. Seven of Miss Murphys students are girls and
5 are boys. In how many different ways can
she seat the 7 girls on the left, then the 5 boys
on the right?
Answer 7! 5!
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Permutations of a set of objects taken from a
larger set Example In a certain lottery game,
four different digits are taken from the digits 1
to 9 to form a 4-digit number. How many different
numbers can be made? Answer 9876
This answer can also be written as 9!/5!
Theorem The number of permutations of r objects
taken from n ( r) objects is
nPr
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Combinations
A collection of objects, in no particular order,
is called a combination.
Example Suppose that there are 5 ingredients
Pepperoni, sausage, green pepper, olive, and
mushroom three are chosen to make a pizza. How
many possible combinations are there?
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Pick 3 items from Pepperoni, Sausage, Green
pepper, Olive, Mushroom.
PSG, PSO, PSM, PGS, PGO, PGM, POS, POG, POM, PMS,
PMG, PMO,
SPG, SPO, SPM, SGO, SGM, SGP, SOP, SOG, SOM, SMP,
SMG, SMO,
GPS, GPO, GPM, GSO, GSP, GSM, GOP, GOS, GOM, GMP,
GMS, GMO,
OPS, OPG, OPM, OSP,OSG, OSM,OGP,OGS,OGM,OMP,
OMS, OMG,
MPS, MPG, MPO,MSP,MSG, MSO,MGP,MGS,MGO,MOP,
MOS,MOG,
We can see that every combination repeats 6
times. Hence we need to divide the answer by 6.
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Theorem The number of combinations of r objects
chosen from n objects, where 0 r n, is
Note Occasionally, nCr is denoted
and read n choose r

• Examples
• 6C2
• 20C12

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Pascals Triangle
0C0 1C0 1C1 2C0 2C1 2C2 3C0 3C1
3C2 3C3 4C0 4C1 4C2 4C3 4C4 . . . . .
1 1 1 1 2 1 1 3 3
1 1 4 6 4 1 1 5
10 10 5 1
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Examples 1. A coin is tossed 7 times, how
many ways are there to get 3 heads and 4
tails?
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2. A candy store has 24 different kinds of
candies. How many ways can you choose 3
different types?
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3. In a class of 16 girls and 14 boys, how many
ways are there to form a committee of 5
girls and 4 boys?
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4. From a standard deck of cards, how many ways
can we make one pair?
Answer (4C2)13
5. From a standard deck of cards, how many ways
can we make a flush poker hand? (a flush
means all 5 cards in the same suit.)
Answer (13C5)4
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Counting by Subtraction
Example 1. In Mrs. ONeills class, there are 20
students. If 16 are girls, how many are boys?
Example 2. In a year of 365 days, if 104 days are
weekends and holidays that you dont work, how
many day do you have to work?
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Example 3. Each student ID is a 5-digit number
(including 00000), how many of these have
duplicate digits?
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Finding probability by subtraction.
Example 1. In a certain raffle, the chance of
winning is only 0.01, what is your chance of
losing?
Example 2. The probability of raining tomorrow
is 25. What is the probability that there is no
rain tomorrow?
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Example 3 If 4 cards are drawn from a standard
deck randomly, what is the probability that they
are from different suits?
Example 4 If 4 cards are drawn from a standard
deck randomly, what is the probability that some
are from the same suit?
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Permutations of not totally distinguishable
objects
Suppose that you have3 indistinguishable yellow
tulips,5 indistinguishable red tulips, 1 purple
tulip, 2 indistinguishable pink tulips.How many
different ways can you arrange them in a row?
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Permutations of not totally distinguishable
objects
Suppose that you have3 indistinguishable yellow
tulips,5 indistinguishable red tulips, 1 purple
tulip, 2 indistinguishable pink tulips.How many
different ways can you arrange them in a row?
Answer Since there are totally 11 tulips we have
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Another example of the same type How many ways
can we rearrange the letters in the word
MISSISSIPPI to get a different string of 11
letters?
Answer Since there are 11 letters totally,
1M, 4Ss, 4 Is, and 2 Ps, there are
!
11



!
2
!
4
!
4
!
1
different arrangements.
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Mathematical Expectations
The expected value of an experiment E is the
average amount one expects to get when the
experiment is repeated a large number of times.
Example Suppose that in a certain game, you
may draw a card from a standard deck. You will be
paid 10 if you get an ace, otherwise you have to
pay 50 cents.If you were to play this game a
large number of times, what will be your average
winnings per game? Should you play this game
often for some extra income?
Answer expected value
and yes, you can play this game more often to get
extra income.
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Note The expected value is not necessarily a
possible outcome in the experiment. For example,
the expected value of rolling a normal six-sided
die is
and we know that it is impossible to roll a 3.5
from a normal die.
This is similar to the statistics that the
average number of children in an American family
is 2.3.
Mathematical expectation is an important
application of probability in gambling, industry,
insurance business, and many other practical
fields. We are going to see several examples
right afterwards.
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Example 1 A U.S. roulette wheel has 38 slots. If
you bet on any single number such as 8, the
casino will pay you 35 to 1. In other words, if
you bet 1 on the number 8, and you win, the
casino will return your 1 and pay you 35. If
you lose, you lose 1. What is the expected
value of this game in the eyes of the casino?
Answer Expected value
This means on average, the casino can get 0.05
for each dollar bet on wheel.
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Example 2 Suppose that an insurance company
has broken down yearly automobile claims for
drivers from age 16 through 21, as shown in the
table below. How much should the company charge
as its average premium in order to break even on
its cost for claims?
Answer Exp 0(0.80) 2000(0.10)
4000(0.05) 6000(0.03) 8000(0.01)
10,000(0.01) 760
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Example 3 Walt, who is a realtor, knows
that if he takes a listing to sell a house, it
will cost him 1,000. However, if he sells the
house, he will receive 6 of the selling price.
If another realtor helps him to sell the house,
he will get 3 of the selling price. If the house
remains unsold after 3 months, he will lose the
listing and receive nothing. Suppose that the
probability for selling 200,000 house are as
follows prob(Walt sells the house)
0.4 prob(another agent sells the house)
0.2 prob(house unsold after 3 months)
0.4 What is Walts expectation if he takes the
listing?
Solution 6 of 200,000 12,000 3 of
200,000 6,000 Expected value 12,0000.4
6,0000.2 1,000
5,000
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The End