Download ON DESK: Agenda

Stats 4 Day 20b
SILENT DO NOW
ON DESK:
Notebook
Agenda:
Ch. 16 Notes
HW Time
DO NOW:
½ sheet on averages
Homework due Monday:
Expected Value Worksheet
Objective
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SWBAT calculate expected value and
define it as an average value.
What is a probability
model?
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Probability Model: a table that shows all
possible outcomes and their
corresponding probabilities
Note: sometimes there are endless
possibilities! This is called a
CONTINUOUS variable. We will be
working with DISCRETE variables
Expected Value
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Take the average of this data set of test scores:
62, 88, 90, 85, 74, 78, 78, 80, 85, 65
If I told you that 15% of students get D’s (say
65), 30% get C’s (75), 40% get Bs (85) and
15% get A’s (95), what would you expect the
average to be?
**Note: make a probability model
***Note: the 3rd column is like saying “how
many out of 100” will get that grade
Expected Value
Formula and Idea
•
Expected Value of a Discrete Random
Variable =
The average value of an unknown or
future data set- the expected
average of repeated trials.
Expected Value
You roll a die – you win $24 if you get
a 6, $12 for a 1 or 2, and nothing for
everything else. How much would
you be willing to pay to play this
game?
Let’s test it out!
Insurance
•
How does insurance work?
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How do they determine the amount of your
deductible?
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You pay a yearly deductible- a certain amount that you must
give the insurance company each year in order to be
covered in case of expenses (ex: break your leg).
They use probability models and expected value!
Also consider life insurance…
You pay a monthly fee for a given payout in case of death,
so if you die within the time period that you are covered for,
the insurance company must pay the agreed upon amount.
They must consider the probability of
• death.
•
Example
•
An insurance company decides to give
a $25,000 payout for death, $5000
payout for disability, and $0 for neither.
If there is a 1/1000 chance of death for
someone your age in your area and a
10/1000 chance of disability, create a
probability model displaying the
possible outcomes.
Betting Games
•
Consider this complicated example using cards (52
cards in a deck)…
I will propose this game to you: You pay $5 to play
this game. If you pull the ace of hearts, I will give
you $100. If you pull any of the other 3 aces, I will
give you $10. If you pull any other of the 12 hearts, I
will give you your $5 back. If you pull any of the
other 36 non-heart cards, you get nothing and I keep
your $5.
• Create a probability model for your EARNINGS
(remember, you give me $5 at the beginning.
• Represent a loss with a negative)
•
Expected Value
1. Set up Probability Model
• 2. Multiply x times P(x) – multiply across
• 3. Add all the xP(x) – add down
•
Expected Value
Example Problem
•
Find the expected value given the
following probability model for a
random variable, x
x
15
30
40
50
P(X=x)
0.2
0.2
0.5
0.1
Expected Value
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Let’s go back to our Betting Game with the
cards from last class:
You pay $5 to play this game. If you pull the ace
of hearts, I will give you $100. If you pull any of
the other 3 aces, I will give you $10. If you pull
any other of the 12 hearts, I will give you your
$5 back. If you pull any of the other 36 nonheart cards, you get nothing and I keep your $5.
•
Find the Expected Value
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Do you think this game is FAIR?
“Fair”
•
A completely “fair” model is one in
which the expected value, E(X),
equals…
• E(X)=0
Create Probability
Model, Calculate E(x)
•
1) find the average number of pops you
have to buy until winning a prize when
there is a 10% chance of winning (you will
buy max 4)
2) find the average amount of money you
will win/lose if you buy pops until you win
a $20 prize if each pop is $2 and the
probability of winning is still 10%
• (you are willing to spend max $8, once
you win you stop buying pop)
•
Expected Value
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Let’s go back to insurance from last class.
An insurance company decides to give a $25,000
payout for death, $5000 payout for disability, and
$0 for neither. If there is a 1/1000 chance of death
for someone your age in your area and a 10/1000
chance of disability, create a probability model
displaying the possible outcomes.
If you are the insurance company, what should you
expect to pay per policy holder? Knowing this, if
you want to make $100 on each customer, what
should you charge each policy holder?
Worksheet
1. Create a probability model
• 2. Calculate the expected value
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Note: When placing bets or playing a
game, if E(X)>0, then you should
play!