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Expected Value
CSCE 115
January 2002
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Probability
 Probability
is determination of the chances
of picking a particular sample from a
known sample.
 Notation:
if A is some event, the
probability of A is
P(A)
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Probability
 Probability
of success (when the events are
equally likely) is
Number of successful outcomes
Number of possible outcomes
 Example: If 1 student is picked at random from
a class of 7 woman and 13 men, what is the
probability that the student is a woman?

P(woman) = 7/(13+7) = 7/20
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Probability
 Non-example:
If you roll two dice and add
the spots the possible outcomes are 2, 3, 4,
5, 6, 7, 8, 9, 10, 11, 12. What is the
probability of rolling a 2?
 Is the following correct?
There is 1 success (getting a two) out of 11
possibilities (2, 3, …, 12) so the probability
is 1/11
 Why?
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Independent events
 Two
events are independent if the way the
first event happens does not affect the way
the second event happens.
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Example: Independent events
 Put
3 red balls and 2 green balls in a bag. Event
1: select a ball at random from the bag and
determine its color. Put the ball back.
Event 2: Select a second ball at random.
Events 1 and 2 are ____________
 Event 3: select a ball at random and set it aside.
Event 4 :select a second ball at random. Events
3 and 4 are ____ _____________
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First fundamental rule:
 The
probability that something does not
happens is 1 - the probability it happens

P(not A) = 1 - P(A)
 Example: The probability of picking a man
from the class of 7 women and 13 men is

1 - (7/20) = 20/20 - (7/20) = 13/20
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Second fundamental rule
 If
two events are independent, the
probability that both A and B happen is
P(A and B) = P(A) * P(B)
 Example: We randomly select a ball from a
bag with 3 red and 2 green balls. We put it
back and draw again. The probability that
both balls are red is
 P(red, red) = (3/5) * (3/5) = 9/25
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Example: Role 2 dice
 Suppose
that we have a red die and a blue die.
We roll and sum. What are the possible
outcomes?
 RB
RB RB RB RB RB
1,1 1,2 1,3 1,4 1,5 1,6
2,2 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4.3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6
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Example: Role 2 dice
 P(2)
= 1/36
P(8) = 5/36
 P(3) = 2/36 = 1/18
P(9) = 4/36 = 1/9
 P(4) = 3/36 = 1/12
P(10) = 3/36 = 1/12
 P(5) = 4/36 = 1/9
P(11) = 2/36 = 1/18
 P(6) = 5/36
P(12) = 1/36

P(7) = 6/36 = 1/6
 Useful
reference
http://www.thewizardofodds.com/game/dice.html Dice Probabilities - The Wizard of Odds
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Probability.
Probability of rolling 2 dice
0.2
0.15
0.1
0.05
0
2
3
4
5
6
7
8
9 10 11 12
Sum of two dice
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A children's game with spinner
The spinner is used to
determine how far you
move. What is the
probability of each
move?
P(2) = 1/10
P(3) = 4/10
P(5) = 1/10
P(6) = 2/10
P(7) = 1/10
P(8)= 1/10
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3
3
2
3
8
5
3
6
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How far, on the average,
do you expect to move
each time you spin?
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A children's game with spinner
 Just
averaging the possible the possible outcomes
(2 + 3 + 5 + 6 + 7 + 8) = 31 = 5.1667
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is _____ correct because the various values are
not equally likely.
 A correct way is
(2 + 3 + 3 + 3 + 3 + 5 + 6 + 6 + 7 + 8) = 46
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10
= 4.6
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A children's game with spinner
 (2
+ 3 + 3 + 3 + 3 + 5 + 6 + 6 + 7 + 8)
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 = (2 +3*4 + 5 + 6*2 + 7 + 8)
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= 2 * 1 + 3 * 4 + 5 * 1 + 6 * 2 + 7 * 1 + 8 * 1
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10
10
10
10
10
 = 2*P(2) + 3*P(3) + 5*P(5) + 6*P(6) + 7*P(7)
+ 8 * P(8)
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Expected value
 Suppose
that a certain experiment X could result
in the values of {a, b, c, …, k} and the
probabilities of these outcomes are P(a), P(b),
P(c), …, P(k). The expected value is
E(X) = a * P(a) + b * P(b) + c * P(c)
+ … + k * P(k)
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Example: Expected value
 Recall
P(2)
the kid's spinner game
= 1/10
P(3) = 4/10
P(5) = 1/10
P(6) = 2/10
P(7) = 1/10
P(8)= 1/10
E(spin) = 2 * .1 + 3 * .4 + 1 * .1
+ 6 * .2 + 7 * .1 + 8 * .1

= 4.6
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Example: Expected value
 Experiment:
You flip a coin and get 1 point
for a head and 0 points for tail
 P(head) = .5,
P(tail) = .5
 E(flip) = 1 * .5 + 0 * .5 = .5
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Example: Expected value
 Experiment:
You roll a die.
 P*(1) = 1/6, P(2) = 1/6, … P(6) = 1/6
 E(roll) = 1*(1/6) + 2*(1/6) + 3*(1/6) +
4*(1/6) + 5*(1/6) +6 *(1/6)
=
1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6
= 21/6 = 3.5
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Example: Expected value
 Experiment:
You roll two dice
 P(2) = 1/36
P(6) = 5/36
P(9) = 4/36
P(3) = 2/36
P(7) = 6/36
P(10) = 3/36
P(4) = 3/36
P(8) = 5/36
P(11) = 2/36
P(5) = 4/36
P(12) = 1/36
 E(two dice) =
2*(1/36) + 3*(2/36) + 4*(3/36) + 5*(4/36)
6*(5/36) + 7*(6/36) + 8*(5/36) + 9*(4/36)
10*(3/36) + 11*(2/36) + 12*(1/36)
 = 252/36 = 7
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Example: Expected value
 John
proposes that the charity bazaar sell tickets
for $2. The player rolls 2 dice. The play wins $12
if the roll is a 2 or a 8. On the average, how much
will the charity win each time a player rolls the
dice?
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Example: Charity bazaar
 Solution
1:
 Outcomes are P(2) = 1/36,
Prize(2) = $12
P(8) = 5/36, Prize(8) = $12
P(other) = 30/36, Prize(other) = $0
E(Prize) = $12*(1/36) + $12*(5/36) +
$0*(30/36) = $72/36 = $2
 Cost of ticket - Expected value of prize
= $2 - $2 = 0
 The charity does not expect to win any money with
this game.
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Example: Charity bazaar
 Solution
2:
 Outcomes are charity wins $2 or loses $10
 P($2) = 30/36 = 5/6
P(-$10) = 1/36 + 5/36 = 6/36 = 1/6
 E(winnings) = $2 * (5/6) + (-$10) * (1/6)
$10/6 + (-$10/6) = 0
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Fair Game
A fair game is a game where the expected value
of winning is 0
 Fair games are highly desirable when play with
your friends
 Fair games are not desirable for organizations
trying to earn money by offering games of chance.
 Casinos would go out of business if they had fair
games.

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Example: Charity bazaar
 John
proposes that the charity bazaar sell tickets for $2.
The player rolls 2 dice. The play wins $12 if the roll is a
2 or a 11. On the average, how much will the charity win
each time a player rolls the dice?
 P(-$10) = 1/36 + 2/36 = 3/36 = 1/12
P($12) = 1- 1/12 = 11/12
 E(winnings) = (-$10) * (1/12) + ($2)*(11/12)
= (-$10 + $22)/12 = $12/12 = $1
 The game is ___ fair. This is desirable for _____.
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Odds
Unfortunately popular slang uses “odds” in at
least 3 different ways.
 It may indicate the payoff if you win a bet
 It may be a synonym for probability

– This is used by the Washington State Lottery
– It is used in many popular articles about odds

It may indicate the ratio of the probability of
winning to the probability of losing
– This is the definition we will use
– This the definition is the one normally sees in math and
statistics books
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Odds of Winning
 Suppose
the probability of winning is p and
probability of losing is q = 1 - p. Then the
odds of winning are p:q.
 We treat p:q as a fraction and normally
multiply and divide both parts to clear of
fractions and to remove common factors.
 The odds of losing are q:p.
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Example: Odds
 In
the revised charity bazaar game, the
probability of a player winning is 1/12.
 The probability of losing is ____
 The odds of winning are 1/12:11/12 or ____
 The odds of losing are ______
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Example: Odds
 You
roll two dice and win if you roll a 9.
 The probability of winning is 4/36
 The probability of losing is ____
 The odds of winning are 4/36:32/36
= 4:32 = ___ : ___
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Example: Odds
 The
odds of winning a game are 5:31. What is the
probability of winning and losing?
 Suppose that you played 5+31 = 36 times. You
would expected to win 5 times and lose 31 times.
 The probability of winning is 5/36.
The probability of losing is 31/36.
 Algebraically, suppose p:q = p:(1-p) = a:b
Treat ":" like it was a "/"
bp = a(1-p) ==> bp = a - ap ==> ap +bp = a
==> (a+b)p = a ==> p = a/(a+b)
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Example: Odds
 The
odds of winning first prize in a raffle
are 1:1999. What is the probability of
winning?
 Suppose that 1+1999 = 2000 tickets are
sold. We would expect to win 1 time out of
2000
 The probability of winning is 1/2000.
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Example: Raffle
 The
prize list for a raffle is
 Prize
Number Value
Odds
New car (Kia)
1
$10,000 1:9999
TV set
10
$300
1:999
Meal for two
20
$50
1:499
 Determine:
– Expected value of a ticket
– Number of tickets sold
– If all of tickets costing $2.50 each are sold and if there is an
addition cost of $2000 for printing and advertising, write a
budget for the sponsors
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