Expected Value

1
Standard MM1D2. d. Use expected value to predict
outcomes.

• Expected Value

2
Expected ValueWhat is it?

• Definition
• The sum of all possible values for a random
  variable each multiplied by its probability of
  occurrence.
• HUH????

3
Lets Think!

• Student members of a service club design a raffle
  to earn money which they will put toward their
  latest service project. The students sell raffle
  tickets for 1 each. They plan to sell 1000
  tickets. The winner of the raffle will win 100.
  All other ticket holders will win 0.

4

• Devon buys one ticket. What is the probability
  she will win the raffle?
• What is the probability Devons single ticket is
  not a winner?
• 3) How much money will the club earn by running
  this raffle?

5

• How could the club design a raffle which would
  earn at least twice the amount of money earned by
  the raffle?
• 5) Derrick is a club member trying to sell
  tickets. He uses the following sales pitch The
  average value of a ticket is 50. Thats a great
  deal! When questioned about this calculation, he
  justifies by saying The value of a ticket is
  either 0 or 100, and the average of 0 and 100
  is 50. Do you agree with Derricks assessment?
  If not, what is the problem?

6
Investigate

• A raffle is held. There are 5000 tickets sold.
• The single first prize ticket is worth 1000.
• There are two second prize tickets worth 100.
• There are five third prize tickets worth 20.
• How much total money will the persons running the
  raffle have to pay out to prize winners?
• 2. How much total money will the raffle earn
  (after prizes are awarded) if tickets are priced
  at 1 each? 2 each?

7

• 3a) What is the probability of a random ticket
  holder winning the first prize?
• 3b) What is the probability of a random ticket
  holder winning a second prize?
• 3c) What is the probability of a random ticket
  holder winning a third prize?
• 3d) What is the probability of a random ticket
  holder winning no prize?

8
Let X represent the amount of dollars won by a
random ticket holder. The possible values of X
are 0, 20, 100 and 1000. Below is an example of a
discrete probability table. Place the
probabilities p(X) which you determined in
question 3 for each value in the appropriate cell
of the table.
x 0 20 100 1000
p(x)
9
Use your table to answer the following

• 4a) What is the probability of a random ticket
  holder winning more than 20?
• 4b) What is the probability of a random ticket
  holder winning at least 20?
• 4c) What is the probability of a random ticket
  holder winning less than 1000?

10
Back to Expected Value!!

• The expected value or mean of a random variable
  is the sum of the probability of each possible
  outcome of the experiment multiplied by the value
  of that outcome.
• Is that the same as the mean you are familiar
  with already?

11
In order to find the expected value of this
raffles payout X, first take each possible
outcome and multiply it by its probability. There
is an added row in the table below for values of
the products of X and p(X).
x 0 20 100 1000
p(x)
Xp(x)
Next, sum the values of Xp(X). This is the
expected value of a random ticket in this raffle.
12

• Expected Value
• 5. How much is the average ticket worth?
• 6. Suppose the group running the raffle set the
  ticket price at exactly the value determined in
  problem 5. Determine how much money the raffle
  would earn after prizes are awarded.
• 7. Use what happened in question 6 to guess what
  would happen in a raffle if the price of a ticket
  was set to be less than the expected value?

13
Example 1 (Book)
14
Example 2 (Book)
15
Example 3 (Book)
16
In PAIRS Solve and Discuss
17
In PAIRS Solve and Discuss