Download Expected Value File

Probability Level 8
• The key idea of probability at Level 8 is investigating
chance situations using probability concepts and
distributions.
• Investigating chance situations using concepts such as
randomness, probabilities of combined events and
mutually exclusive events, independence, conditional
probabilities and expected values and standard
deviations of discrete random variables, and
probability distributions including the Poisson,
binomial and normal distributions.
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Chapter 1
Random variables
• A random variable, X, is a numerical measure of
the outcomes of an experiment
• Random variables can be discrete or continuous
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Chapter 1
Random variables
• Discrete random variables have a countable number of
outcomes
– Examples: Dead/alive, treatment/placebo, dice, counts, etc.
• Continuous random variables have an infinite
continuum of possible values.
– Examples: blood pressure, weight, the speed of a car, the real
numbers from 1 to 6.
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Chapter 1
Expected Value
The Expected Value of a random variable is a kind of
theoretical average that it should take. It is an
extremely useful concept for good decision making!
The symbol for Expected Value is E(X)
The Expected Value is often referred to as the
population mean (symbol µ )
Expected Value = E(X) = µ
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Expected Value
• It is an extremely useful concept for good
decision making and is used in
• Games and Gambling
• Investments
• Business and Insurance
12-5-5
© 2008 Pearson Addison-Wesley. All rights reserved
Expected Value
• It is an extremely useful concept for good
decision making and is used in
• Games and Gambling
• Investments
• Business and Insurance
12-5-6
© 2008 Pearson Addison-Wesley. All rights reserved
To work out the expected value of a
random variable from a probability table
• MULTIPLY each possible value of X by its
probability
• Then ADD these products
Discrete case:
E( X ) 
 x p(x )
i
i
all x
12-5-7
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Expected Value example
Children in third grade were surveyed and told to pick
the number of hours that they play electronic games
each day. The probability distribution is given below.
x (hrs)
0
1
2
3
P(x)
0.3
0.4
0.2
0.1
Calculate a “weighted average” by multiplying
each possible time value by its probability and then
adding the products.
12-5-8
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Expected Value
0(.3)  1(.4)  2(.2)  3(.1)  1.1
1.1 hours is the expected value (or the mathematical
expectation) of the quantity of time spent playing
electronic games.
12-5-9
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Example: Finding Expected Value
Find the expected number of boys for a three-child
family. Assume girls and boys are equally likely.
Solution
S = {ggg, ggb, gbg,
bgg, gbb, bgb, bbg,
bbb}
The probability
distribution is on
the right.
# Boys Probability
x
P(x)
0
1/8
1
3/8
2
3/8
3
1/8
Product
x  P( x)
0
3/8
6/8
3/8
12-5-10
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Example: Finding Expected Value
Solution (continued)
The expected value is the sum of the third column:
3 6 3 12
0   
8 8 8 8
3
  1.5.
2
So the expected number of boys is 1.5. In context this
means that if a large number of three child families were
studied the “average” number of girls in these families
would be very close to1.5.
12-5-11
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Example: Finding Expected Winnings
A player pays $3 to play the following game: He rolls
a die and receives $7 if he tosses a 6 and $1 for
anything else. Find the player’s expected net winnings
for the game.
Die Outcome Payoff Net P(x)
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Example: Finding Expected Winnings
Solution
The information for the game is displayed below.
Die Outcome Payoff Net P(x) x  P ( x )
1, 2, 3, 4, or 5
$1
–$2 5/6
6
$7
$4
–$10/6
1/6
$4/6
Expected value: E(x) = –$6/6 = –$1.00
12-5-13
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Games and Gambling
A game in which the expected net winnings
are zero is called a fair game. A game with
negative expected winnings is unfair against
the player. A game with positive expected net
winnings is unfair in favor of the player.
12-5-14
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Example: Finding the Cost for a Fair
Game
What should the game in the previous example
cost so that it is a fair game?
Solution
Because the cost of $3 resulted in a net loss of $1,
we can conclude that the $3 cost was $1 too high. A
fair cost to play the game would be $3 – $1 = $2.
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Investments
Expected value can be a useful tool for
evaluating investment opportunities.
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Example: Expected Investment Profits
Mark is going to invest in the stock of one of the two
companies below. Based on his research, a $6000
investment could give the following returns.
Company ABC
Company PDQ
Profit or Probability Profit or Probability
Loss x
P(x)
Loss x
P(x)
–$400
.2
$600
.8
$800
.5
1000
.2
$1300
.3
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Example: Expected Investment Profits
Find the expected profit (or loss) for each of the
two stocks.
Solution
ABC: –$400(.2) + $800(.5) + $1300(.3) = $710
PDQ: $600(.8) + $1000(.2) = $680
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Business and Insurance
Expected value can be used to help make
decisions in various areas of business,
including insurance.
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Example: Expected Lumber Revenue
A lumber wholesaler is planning on purchasing a
load of lumber. He calculates that the probabilities
of reselling the load for $9500, $9000, or $8500 are
.25, .60, and .15, respectfully. In order to ensure an
expected profit of at least $2500, how much can he
afford to pay for the load?
12-5-20
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Example: Expected Lumber Revenue
Solution
The expected revenue from sales can be found below.
Income x
P(x)
x  P( x)
$9500
.25
$2375
$9000
.60
$5400
$8500
.15
$1275
Expected revenue: E(x) = $9050
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Example: Expected Lumber Revenue
Solution (continued)
profit = revenue – cost or cost = profit – revenue
To have an expected profit of $2500, he can pay up to
$9050 – $2500 = $6550.
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