11.8 Expected Value

11.8 Expected Value
Ks OBJECTIVES
a O EE оо М Me И НН НН
-
1 Compute expected value.
| Use expected value to solve
Ao pues jue mG eR БЫ ББК EM EIN BN ES EE И Wm wd WE
applied problems.
: Use expected value to
determine the average payoff
or loss in a game of chance.
| Compute expected value.
"очен он к он но EEN EN EN WE ES ESE EOF MS DON
SECTION 11.8 Expected Value 659
EI we mew mee eee ee A RC GN ME ES BEN SEN HE BEN НН БЫ Бонн НЫ Бы НЫ EGE BSN НЫ БН BOO НЮ ОМК DN WD DOR НМ МЫ BM ee ME Кан GN НН НН НН Em ee MN EC EE но Ня ED ER НН НЫ НН Ee Re MM Но вая mad ses me me
Expected Value
. Would you be willing to spend $50 a year for an insurance
. policy that pays $200,000 if you become too ill to
.¥ continue your education? It is unlikely that this will
Insurance companies make money by
compensating us for events that have a low
© probability. If one in every 5000
… students needs to quit college due to
“ serious illness, the probability of this
event is =000- Multiplying the amount
of the claim, $200,000, by its probability,
в i, tells the insurance company what
to expect to pay out on average for each policy:
1
Ea
$200,000 X 5000 7 $40.
Amount of Probability of
the claim paying the claim
Over the long run, the insurance company can expect to pay $40 for each policy it
sells. By selling the policy for $50, the expected profit 1s $10 per policy. If 400,000 stu-
dents choose to take out this insurance, the company. can expect to make
400,000 Xx $10, or $4,000,000.
Expected value is a mathematical way to use probabilities to determine what
to expect in various situations over the long run. Expected value is used to
determine premiums on insurance policies, weigh the risks versus the benefits of
alternatives in business ventures, and indicate to a player of any game of chance
what will happen if the game is played repeatedly.
The standard way to find expected value is to multiply each possible outcome
by its probability, and then add these products. We use the letter £ to represent
expected value.
Computing Expected Value
Find the expected value for the outcome of the roll of a fair die.
Solution The outcomes are 1, 2, 3, 4, 5, and 6, each with a probability of Е The
expected value, E, is computed by multiplying each outcome by its probability
and then adding these products.
1 1 1 1 1 1
.— + .— *— + . — *— + . —
des ar EE 6
1594344455 |
- 6 Te 699
E
The expected value of the roll of a fair die is 3.5. This means that if the die is rolled
repeatedly, there is an average of 3.5 dots per roll over the long run. This expected
value cannot occur on a single roll of the die. However, it is a long-run average of
the various outcomes that can occur when a fair die 1s rolled.
It is equally probable that a pointer will land on any one of four regions,
numbered 1 through 4. Find the expected value for where the pointer will
stop.
660 CHAPTER 11 Counting Methods and Probability Theory
TABLE 11.8 Outcomes and
Probabilities for the Number of Girls in
a Three-Child Family
Outcome:
Number of Girls Probability
0
ol
1
o
2
cop cols
3
TABLE 11.9
Number of Heads | Probability
fe a >) — E
S| 5s [Sle [5s | 5
Use expected value to solve
applied problems.
TABLE 11.10 Probabilities for
Auto Claims
| Amouni of
Claim (to the
nearest $2000) Probability
$0 0.70
$2000 0.15
$4000 0.08
$6000 0.05
$8000 0.01
$10,000 0.01
Computing Expected Value
Find the expected value for the number of girls for a family with three children.
Solution A family with three children can have 0, 1,2, or 3 girls. There are eight
ways these outcomes can occur.
One way
No girls : Boy Boy Boy
One girl: Girl Boy Boy, Boy Girl Boy, Boy Boy Girl ee
Two girls : Girl Girl Boy, Girl Boy Girl, Boy Girl Girl — "tree ways
Three girls : Girl Girl Girl
Three ways
One way
Table 11.8 shows the probabilities for 0,1, 2, and 3 girls.
The expected value, E, is computed by multiplying each outcome by its
probability and then adding these products.
1 3 3 Ps 3 +643 12 3
E == 0: сев Те НЕ же Ве = — = — = 1.5
3 8 8 8 8 8 2
The expected value 1s 1.5. This means that if we record the number of girls in many
different three-child families, the average number of girls for all these families will
be 1.5. In a three-child family, half the children are expected to be girls, so the
expected value of 1.5 1s consistent with this observation.
A fair coin 1s tossed four times in succession. Table 11.9 shows the
probabilities for the different number of heads that can arise. Find the
expected value for the number of heads.
Applications of Expected Value
Empirical probabilities can be determined in many situations by examining what
has occurred in the past. For example, an insurance company can tally various claim
amounts over many years. If 15% of these amounts are for a $2000 claim, then the
probability of this claim amount 1s 0.15. By studying sales of similar houses in a
particular area, a realtor can determine the probability that he or she will sell a
listed house, another agent will sell the house, or the listed house will remain unsold.
Once probabilities have been assigned to all possible outcomes, expected value can
indicate what 1s expected to happen in the long run. These ideas are illustrated in
Examples 3 and 4.
Determining an Insurance Premium
An automobile msurance company has determined the probabilities for various
claim amounts for drivers ages 16 through 21, shown in Table 11.10.
a. Calculate the expected value and describe what this means in practical terms.
b. How much should the company charge as an average premium so that it does
not lose or gain money on its claim costs?
Solution
a. The expected value, FE, is computed by multiplying each outcome by its
probability and then adding these products.
E = $0(0.70) + $2000(0.15) + $4000(0.08) + $6000(0.05)
+ $8000(0.01) + $10,000(0.01)
= $0 + $300 + $320 + $300 + $80 + $100
= $1100
The expected value is $1100. This means that in the long run the average cost
of a claim is $1100. The insurance company should expect to pay $1100 per
car insured to people in the 16-21 age group.
THE REALTOR'S SUMMARY
HEET
My Cost: $5000
My Possible Income: | |
I sell house: $30,000
Another agent sells
house: $15,000
House unsold after
4 months: $0
The Probabilities:
I sell house: 0.3
Another agent sells
house: 0.2
House unsold after
4 months: | 0.5
My Bottom Line:
I take the listing only 1f 1
anticipate earning at least $6000.
SECTION 11.8 Expected Value 661
b. At the very least, the amount that the company should charge as an average
premium for each person in the 16-21 age group is $1100. In this way, it will
not lose or gain money on its claims costs. It’s quite probable that the
company will charge more, moving from break-even to profit.
Work Example 3 again if the probabilities for claims of $0 and $10,000 are
“ reversed. Thus, the probability of a $0 claim is 0.01 and the probability of a
$10,000 claim is 0.70.
E SA A A WO AN A m БАШ Gr kl | PEN MAE ME Mm De tb! Ме Мед DE pe dar die a ma Em A TE We AE ee De WD MI Dm A ee SW Te A A DO RR nd EA meh ni ii
Business decisions are interpreted in terms of dollars and cents. In these
situations, expected value is calculated by multiplying the gain or loss for each
possible outcome by its probability. The sum of these products is the expected value.
Expectation in a Business Decision
You are a realtor considering listing a $500,000 house. The cost of advertising and
providing food for other realtors during open showings is anticipated to cost you
$5000. The house is quite unusual and you are given a four-month listing. It the
house is unsold after four months, you lose the listing and receive nothing. You
anticipate that the probability you sell your own listed house is 0.3, the probability
that another agent sells your listing is 0.2, and the probability that the house is
unsold after 4 months is 0.5. If you sell your own listed house, the commission 15 а
hefty $30,000. If another realtor sells your listing, the commission is $15,000. The
bottom line: You will not take the listing unless you anticipate earning at least
$6000. Should you list the house?
Solution Shown in the margin is a summary of the amounts of money and
probabilities that will determine your decision. The expected value in this situation
is the sum of each income possibility times its probability. The expected value
represents the amount you can anticipate earning if you take the listing. If the
expected value is not at least $6000, you should not list the house.
The possible incomes listed in the margin, $30,000, $15,000, and $0, do not take
into account your $5000 costs. Because of these costs, each amount needs to be
reduced by $5000. Thus, you can gain $30,000 — $5000, or $25,000, or you can gain
$15,000 — $5000, or $10,000. Because $0 — $5000 = —$5000, you can also lose
$5000. Table 11.11 summarizes possible outcomes if you take the listing, and their
respective probabilities.
TABLE 11.11 Gains, Losses, and Probabilities for Listing a $500,000 House
Outcome | Gain or Loss Probability
Sells house $25,000 0.3
Another agent sells house $10,000 0.2
House doesn't sell —$5000 0.5
The expected value, E, is computed by multiplying each gain or loss in Table 11.11
by its probability and then adding these results.
Е = $25,000(0.3) + $10,000(0.2) + (—$5000)(0.5)
= $7500 + $2000 + (—$2500) = $7000
You can expect to earn $7000 by listing the house. Because the expected value
exceeds $6000, you should list the house.
662 CHAPTER 11 Counting Methods and Probability Theory
Use expected value to
determine the average payoff
or loss in a game of chance.
AWAY
probabilities and payoffs that
result in negative expected
values. This means that players
will lose money in the long run.
With no clocks on the walls or
windows to look out of, casinos
bet that players will forget about
time and play longer. This brings
casino owners closer to what
they can expect to earn, while
negative expected values sneak
up on losing patrons. (No
wonder they provide free
beverages!)
A Potpourri of Gambling-
Related Data
* Population of Las Vegas:
535,000
* Slot Machines in Las
Vegas: 150,000
* Average Earnings per Slot
Machine per Year: $100,000
* Average Amount Lost per
Hour in Las Vegas Casinos:
$696,000
* Average Amount an
American Adult Loses
Gambling per Year: $350
* Average Amount a Nevada
Resident Loses Gambling
per Year: $1000
Source: Paul Grobman, Vital
Statistics, Plume, 2005.
get ONE OND NE NR ONE ONS MR oS RR osm em em O O вн Sm O он он нон но но он EE EE EN EE EN OO = me Re me кн mm mm ME EE ME о EE ME MEN ME ME EE RE Re fe es es ee Ee в новы о EN MN MN ME ME но кн оно ны кн ны вн вы вы ок pl O we O ши нож ни ME вн ош me кн ош к о оно оно о
Rom wm mm em mE RE RE WE EE MS EE ee ен we mm mm Rm mm БЫ жи к WN NN и пы RE ни вн EN NR SS RR Bm Mm ши WN MN ME EN EN EN но кн вы кн на оны кн кв mm em ож ож мн жи mm mm mm mm mm жа Em mm Em ны Me Me EE Sm а шв ны кн кн жи вн mm mm mm mm mm mm mm Em Em Mm Em Em em A me em ee ое == == == =
The SAT 1s a multiple-choice test. Each question has five possible
answers. The test taker must select one answer for each question or not
answer the question. One point is awarded for each correct response and
+ point is subtracted for each wrong answer. No points are added or
subtracted for answers left blank. Table 11.12 summarizes the information
for the outcomes of a random guess on an SAT question. Find the
expected point value of a random guess. Is there anything to gain or lose
on average by guessing? Explain your answer.
TABLE 11.12 Gains and Losses for Guessing on the SAT
Outcome Gain or Loss Probability
Guess correctly 1 Е
T 1 4
Guess incorrectly 4 =
Expected Value and Games of Chance
Expected value can be interpreted as the average payoff in a contest or game when
either 1s played a large number of times. To find the expected value of a game,
multiply the gain or loss for each possible outcome by its probability. Then add the
products.
A game is played using one die. If the die is rolled and shows 1, 2, or 3, the player
wins nothing. If the die shows 4 or 5, the player wins $3. If the die shows 6, the
player wins $9. If there is a charge of $1 to play the game, what is the game’s
expected value? Describe what this means in practical terms.
Solution Because there is a charge of $1 to play the game, a player who wins $9
gains $9 — $1, or $8. A player who wins $3 gains $3 — $1, or $2. If the player gets $0,
there is a loss of $1 because $0 — $1 = —$1. The outcomes for the die, with their
respective gains, losses, and probabilities, are summarized in Table 11.13.
TABLE 11.13 Gains, Losses, and Probabilities in a Game of Chance
Outcome Gain or Loss Probability
1,2,0r 3 —$1 я
4 or 5 $2 $
6 $8 :
Expected value, E, is computed by multiplying each gain or loss in Table 11.13 by its
probability and then adding these results.
Е = с50)(2) + (5) #65)
— —$3 + $4 + $8 _ 59 _
= 6 = 5 = $1.50
The expected value is $1.50. This means that in the long run, a player can expect to
win an average of $1.50 for each game played. However, this does not mean that the
player will win $1.50 on any single game. It does mean that if the game is played
repeatedly, then, in the long run, the player should expect to win about $1.50 per
play on the average. If 1000 games are played, one could expect to win $1500.
However, if only three games are played, one’s net winnings can range between —$3
and $24, even though the expected winnings are $1.50(3), or $4.50.
I
E
i
E
|
i
i
i
I
i
i
i
i
i
1
I
I
i
I
i
1
!
i
I
I
1
i
1
1
I
I
i
i
I
E
i
E
i
i
i
i
1
1
I
i
i
i
i
i
1
i
1
i
1
i
i
i
I
I
i
i
i
%
FIGURE 11.14 A U.S. roulette
wheel
BLITZER BONUS
HOW TO WIN AT
ROULETTE
A player may win or lose at
roulette, but in the long run the
casino always wins. Casinos make
an average of three cents on
every dollar spent by gamblers.
There is a way to win at roulette
in the long run: Own the casino.
"a
Ey e ME O O O e e e O O БЫ БЫ БЫ ББ EN O E O e e me ee OO O AS O EN O и нон НН НЫ Ee SO O Em SE EE DE O O mE = но = == ==
SECTION 11.8 Expected Value 663
A charity is holding a raffle and sells 1000 raffle tickets for $2 each. One
of the tickets will be selected to win a grand prize of $1000. Two other
tickets will be selected to win consolation prizes of $50 each. Fill in the
missing column in Table 11.14. Then find the expected value if you buy
one raffle ticket. Describe what this means in practical terms. What can
you expect to happen if you purchase five tickets?
TABLE 11.14 Gains, Losses, and Probabilities in a Raffle
Outcome Gain or Loss Probability
Win Grand Prize тот
Win Consolation Prize 100
Win Nothing т
а бы le А TE ни цы ей ны Ц AR RE ET EN ЦН СВР САБЫ ЧЕМ el De LL i ЕП НА
Unlike the game in Example 5, games in gambling casinos are set up so that
players will lose in the long run. These games have negative expected values. Such a
game is roulette, French for “little wheel.” We first saw the roulette wheel in Section
11.7. It is shown again in Figure 11.14. Recall that the wheel has 38 numbered slots
(1 through 36, 0, and 00). In each play of the game, the dealer spins the wheel and a
small ball in opposite directions. The ball is equally likely to come to rest in any one
of the slots, which are colored black, red, or green. Gamblers can place a number of
different bets in roulette. Example 6 illustrates one gambling option.
Expected Value and Roulette
One way to bet in roulette is to place $1 on a single number. If the ball lands on
that number, you are awarded $35 and get to keep the $1 that you paid to play the
game. If the ball lands on any one of the other 37 slots, you are awarded nothing
and the $1 that you bet is collected. Find the expected value for playing roulette if
you bet $1 on number 20. Describe what this means.
Solution Table 11.15 contains the two outcomes of interest: the ball landing on
your number, 20, and the ball landing elsewhere (in any one of the other 37 slots). The
outcomes, their respective gains, losses, and probabilities, are summarized in the table.
TABLE 11.15 Playing One Number with a 35 to 1 Payoff in Roulette
Outcome Gain or Loss Probability
Ball lands on 20 $35 +
Ball does not land on 20 —$1 E
E
Expected value, E, is computed by multiplying each gain or loss in Table 11.15 by its
probability and then adding these results.
— ea 1 6/37) $35 $37 _ 2
в = $5(55) + < (5) a AAA
The expected value is approximately — $0.05. This means that in the long run, a player
can expect to lose about 5¢ for each game played. If 2000 games are played, one could
expect to lose $100.
In the game of one-spot keno, a card is purchased for $1. It allows a player
to choose one number from 1 to 80. A dealer then chooses twenty
numbers at random. If the player’s number is among those chosen, the
player is paid $3.20, but does not get to keep the $1 paid to play the game.
Find the expected value of a $1 bet. Describe what this means.
664 CHAPTER 11 Counting Methods and Probability Theory
Practice and Application Exercises 5. An architect is considering bidding for the design of a new
| : museum. The cost of drawing plans and submitting a model
In Exercises 1-2, the numbers that each pointer can land on and is $10,000. The probability of being awarded the bid is 0.1,
their respective probabilities are shown. Compute the expected and anticipated profits are $100,000, resulting in a possible
value for the number on which each pointer lands. gain of this amount minus the $10,000 cost for plans and a
1. Outcome | Probability model. What is the expected value 1 this situation? Describe
ES ZA what this value means.
1 3 6. A construction company is planning to bid on a building
9 + contract. The bid costs the company $1500. The probability
: _ that the bid is accepted is =. If the bid is accepted, the
3 4 company will make $40,000 minus the cost of the bid. Find
the expected value in this situation. Describe what this value
means.
7. It is estimated that there are 27 deaths for every 10 million
people who use airplanes. A company that sells flight
insurance provides $100,000 in case of death in a plane crash.
у Outcome | Probability A policy can be purchased for $1. Calculate the expected value
nn bt and thereby determine how much the insurance company can
1 = make over the long run for each policy that it sells.
> ia + tes 8. A 25-year-old can purchase a one-year life insurance policy
| — for $10,000 at a cost of $100. Past history indicates that the
3 ; probability of a person dying at age 25 is 0.002. Determine
4 1 TEA the company's expected gain per policy.
Exercises 9-10 are related to the SAT, described in Check Point 4
on page 602.
9. Suppose that you can eliminate one of the possible five
answers. Modify the two probabilities shown in the final
The tables in Exercises 3—4 show claims and their probabilities for column in Table 11.12 on page 662 by finding the
an insurance company. probabilities of guessing correctly and guessing incorrectly
a. Calculate the expected value and describe what this means under these circumstances. What is the expected point value
in practical terms. of a random guess? Is it advantageous to guess under these
b. How much should the company charge as an average ~~ circumstances?
premium so that it breaks even on its claim costs? 10. Suppose that you can eliminate two of the possible five
c. How much should the company charge to make a profit of answers. Modify the two probabilities shown in the final
$50 per policy? column in Table 11.12 on page 662 by finding the
3. PROBABILITIES FOR HOMEOWNERS' probabilities of guessing correctly and guessing incorrectly
INSURANCE CLAIMS under these circumstances. What is the expected point value
Amount of Claim (to the of a random guess? Is it advantageous to guess under these
nearest $50,000) Probability circumstances?
E 7 0.65 11. A store specializing in mountain bikes is to open in one of
. two malls. If the first mall is selected, the store anticipates a
$50,000 0.20 | :
. ————— yearly profit of $300,000 if successful and a yearly loss of
_ $100,000 010° $100,000 otherwise. The probability of success is +. If the
$150,000 0.03 second mall 1s selected, it is estimated that the yearly profit
$200,000 | … 001 | will be $200,000 if successful; otherwise, the annual loss will
$250,000 EA 5 be $60,000. The probability of success at the second mall is 3.
Which mall should be chosen in order to maximize the
4. PROBABILITIES FOR MEDICAL INSURANCE expected profit?
CLAIMS 12. An oil company is considering two sites on which to drill,
Amount of Claim (to the described as follows:
teca | aah Site A: Profit if oil is found: $80 million
$0 0.70 | Loss if no oil is found: $10 million
$20,000 0.20 Probability of finding oil: 0.2
PO à PE Site B: Profit if oil is found: $120 million
anne a $60,000 0.02 Loss if no oil is found: $18 million
$80,000 0.01 Probability of finding oil: 0.1
pL00.000 Sle) Which site has the larger expected profit? By how much?
13.
14.
In a product liability case, a company can settle out of court
for a loss of $350,000, or go to trial, losing $700,000 if found
guilty and nothing if found not guilty. Lawyers for the
company estimate the probability of a not-guilty verdict to
be 0.8.
a. Find the expected value of the amount the company can
lose by taking the case to court.
b. Should the company settle out of court?
A service that repairs air conditioners sells maintenance
agreements for $80 a year. The average cost for repairing an
air conditioner is $350 and 1 in every 100 people who
purchase maintenance agreements have air conditioners
that require repair. Find the service’s expected profit per
maintenance agreement.
Exercises 15-19 involve computing expected values in games of
chance.
15.
16.
17.
18.
19.
Yr
vy
20.
21.
22.
a a WF PS BE
в ое п еб Fo ER РО Ee
¡Line in Mar
ET
-
E 3
A game is played using one die. If the die is rolled and shows
1, the player wins $5. If the die shows any number other than
1, the player wins nothing. If there is a charge of $1 to play
the game, what is the game’s expected value? What does this
value mean?
A game is played using one die. If the die is rolled and shows
1, the player wins $1; if 2, the player wins $2; if 3, the player
wins $3. If the die shows 4, 5, or 6, the player wins nothing. If
there is a charge of $1.25 to play the game, what is the game’s
expected value? What does this value mean?
Another option in a roulette game (see Example 6 on page
663) is to bet $1 on red. (There are 18 red compartments,
18 black compartments, and 2 compartments that are neither
red nor black.) If the ball lands on red, you get to keep the $1
that you paid to play the game and you are awarded $1. If
the ball lands elsewhere, you are awarded nothing and the $1
that you bet is collected. Find the expected value for playing
roulette if you bet $1 on red. Describe what this number
means.
The spinner on a wheel of fortune can land with an equal
chance on any one of ten regions. Three regions are red, four
are blue, two are yellow, and one is green. A player wins $4 if
the spinner stops on red and $2 if it stops on green. The
player loses $2 if it stops on blue and $3 if it stops on yellow.
What is the expected value? What does this mean if the
game is played ten times?
For many years, organized crime ran a numbers game that
is now run legally by many state governments. The player
selects a three-digit number from 000 to 999. There are
1000 such numbers. A bet of $1 is placed on a number, say
number 115. If the number is selected, the player wins $500.
If any other number is selected, the player wins nothing.
Find the expected value for this game and describe what
this means.
i
E
a
: д 2
BT A, Y “e y e pe
вела 5
вы
в
i
What does the expected value for the outcome of the roll of
a fair die represent?
Explain how to find the expected value for the number of
girls for a family with two children. What is the expected
value?
How do insurance companies use expected value to
determine what to charge for a policy?
SECTION 11.8 Expected Value 665
23. Describe a situation in which a business can use expected
value.
24. If the expected value of a game is negative, what does this
mean? Also describe the meaning of a positive and a zero
expected value.
25. The expected value for purchasing a ticket in a raffle is
—$0.75. Describe what this means. Will a person who
purchases a ticket lose $0.75?
Critical Thinking Exercises
Make Sense? In Exercises 26-29, determine whether each statement
makes sense or does not make sense, and explain your reasoning.
26. Ifound the expected value for the number of boys for a family
with five children to be 2.5.1 must have made an error because
a family with 2.5 boys cannot occur.
27. Here’s my dilemma: I can accept a $1000 bill or play a dice
game ten times. For each roll of the single die,
® I win $500 for rolling 1 or 2.
* J win $200 for rolling 3.
e lose $300 for rolling 4, 5, or 6.
Based on expected value, I should accept the $1000
bill.
28. Гуе 1051 а fortune playing roulette, so I'm bound to reduce
my losses if I play the game a little longer.
29. My expected value in a state lottery game is $7.50.
30. A popular state lottery is the 5/35 lottery, played in Arizona,
Connecticut, Illinois, Iowa, Kentucky, Maine, Massachusetts,
New Hampshire, South Dakota, and Vermont. In Arizona's
version of the game, prizes are set: First prize is $50,000,
second prize is $500, and third prize is $5. To win first prize,
you must select all five of the winning numbers, numbered
from 1 to 35. Second prize is awarded to players who select
any four of the five winning numbers, and third prize is
— awarded to players who select any three of the winning
numbers. The cost to purchase a lottery ticket is $1. Find the
expected value of Arizona's “Fantasy Five” game, and
describe what this means in terms of buying a lottery ticket
over the long run.
31. Refer to the probabilities of dying at any given age on page
630 to solve this exercise. A 20-year-old woman wants to
purchase a $200,000 one-year life insurance policy. What
should the insurance company charge the woman for the
policy if it wants an expected profit of $60?
Group Exercise
32. This activity is a group research project intended for people
interested in games of chance at casinos. The research should
culminate in a seminar on games of chance and their
expected values. The seminar is intended to last about
30 minutes and should result in an interesting and
informative presentation made to the entire class.
Each member of the group should research a game avail-
able at a typical casino. Describe the game to the class and
compute its expected value. After each member has done
this, so that class members now have an idea of those games
with the greatest and smallest house advantages, a final
group member might want to research and present ways for
currently treating people whose addiction to these games
has caused their lives to swirl out of control.
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement