Transcript Expected Value - Wando High School

You Bet Your Life
- So to Speak
Chapter 16
Life Insurance
• An insurance company offers a “death and
disability” policy that pays $10,000 when you die
or $5000 if you are permanently disabled. It
charges a premium of only $50 a year for this
benefit.
• Is the company likely to make a profit selling such
a plan?
• The company looks at the probability that its
clients will die or be disabled in any year.
• This actuarial information helps the company
calculate the expected value of this policy.
We’ll want to build a probability model in order
to answer the questions about the insurance
company’s risk.
First we need to define
a few terms.
Random Variables
• Random variable - a numeric value which is
based on the outcome of a random event.
• Represent with a capital letter, like X
• Use the lowercase to denote any particular
value the Random Variable have, like x
Note: The most common letters are X, Y, and Z.
But be cautious: If you see any capital letter, it
just might denote a random variable.
Random Variables
•
•
•
•
For the insurance company, x can be
$10,000 (if you die that year)
$5000 (if you are disabled)
$0 (if neither occurs).
Probability Model
• The Probability Model for the Random
Variable is the collection of all the possible
values and their probabilities.
Suppose, the death rate in any year is 1 out of
every 1000 people, and that another 2 out of
1000 suffer some kind of disability. Then we
can display the probability model for this
insurance policy in a table like this:
Expected Value (Center)
• The expected value is a parameter
• In fact, it’s the mean.
• We’ll signify this with the notation  (for
population mean) or E(X) for expected value.
What Can the Insurance Company Expect?
Imagine the company insures exactly 1000 people.
Further imagine that, in a perfect “probability world,”:
1 of the policyholders dies - $10,000
2 are disabled - $5000 each
997 survive the year unscathed - $0 each
$20000
A total of $20,000 or an average of
= $20 per policy.
1000
Since it is charging people $50 for the policy, the
company expects to make a profit of $30 per customer.
We can’t predict what will happen during any given
year, but we can say what we expect to happen. To
do this, we (or, rather, the insurance company) need
the probability model.
How did we come up with $20 as the expected
value of a policy payout?
We imagined that we had exactly 1000 clients. Of
those, we imagined exactly 1 died and 2 were
disabled, corresponding to the probabilities.
Our average payout is:
10000(1)  5000(2)  0(997)
  E( X ) 
 $20 per policy
1000
Expected Value
A $20 bill, two $10 bills, three $5 bills and four $1
bills are placed in a bag. If a bill is chosen at
random, what is the expected value for the
amount chosen?
Outcome
Probability
$20
1/10
$10
2/10
$5
3/10
$1
4/10
1
2
3
4
$20*  10*  5*  1* 
10
10
10
10
$2  $2  $1.5  $0.4  $5.9
The expected value is $5.90
In a game you flip a coin twice, and record the
number of heads that occur. You get 10 points for
2 heads, zero points for 1 head, and 5 points for no
heads. What is the expected value for the number
Outcome
Probability
of points you’ll win per turn?
2 Heads
1/4
1 Head
1/2
No Heads
1/4
1
1
1
10*  0*  5* 
4
2
4
2.5  0  1.25  3.75
The expected value is 3.75
There is an equally likely chance that a
falling dart will land anywhere on the rug
below.
The following system is used to find the
number of points the player wins.
What is the expected value for the number
of points won?
Black = 40 points
Gray = 20 points
White = 0 points
Outcome
Value
Probability
Black
40
6/15 = 2/5
Gray
20
6/15 = 2/5
White
0
3/15 = 1/5
2
2
1
E ( x)  40*  20*  0* 
5
5
5
16  8  24
A mysterious card-playing squirrel (pictured) offers
you the opportunity to join in his game. The rules
are: To play you must pay him $2. If you pick a
spade from a shuffled pack, you win $9. Find the
expected value you win (or lose) per game.
Outcome
Value
Probability
Spade
$9- $2 = $7
1/4
Other
-$2
3/4
1
3 $7 6
$7 *  ($2) * 

 $0.25
4
4 4
4
A dice game involves rolling 2 dice. If you roll a 2,
3, 4, 10, 11, or a 12 you win $5. If you roll a 5, 6, 7,
8, or 9 you lose $5. Find the expected value you
win (or lose) per game.
Outcome
(Sum)
Value
Probability
2, 3, 4, 10,
11, 12
$5
6/11
5, 6, 7, 8, 9
-$5
5/11
6
5 $30 $25 $5
$5*  ($5) * 


11
11 11
11
11
Recall:
Population Mean = Expected Value
First Center, Now Spread . . .
• Of course, this expected value (or mean) is not what
actually happens to any particular policyholder. No
individual policy actually costs the company $20. We
are dealing with random events, so some policyholders
receive big payouts, others nothing. Because the
insurance company must anticipate this variability, it
needs to know the standard deviation of the random
variable.
• For data, we calculated the standard deviation by first
computing the deviation from the mean and squaring
it.
• We do that with (discrete) random variables as well.
First, we find the deviation of each payout
from the mean (expected value):
Here are the formulas for what we just did.
Because these are parameters of our probability
model, the variance and standard deviation can
also be written as  and  .
2
You should recognize both kinds of notation.
Skills
Classwork & Homework
Classwork: 16.1 WS Probability
Homework: pp. 363 – 364 (1 – 8)