Transcript What Do You Expect

What Do You Expect
D. Brooke Hill & Rose Sinicrope
East Carolina University
1530-1645, Thursday, 31 October 2013
NCCTM 43rd Annual Conference
Auditorium III, Joseph S. Koury Convention Center
Greensboro, NC
Roll two regular hexahedral dice.
What is the sum?
What do you expect will happen?
On the average, what percent of the time, will
you get
2?
3?
:
What sum can you expect?
PROBABILITY HISTOGRAM
2
3
4
5
6
7
8
9
Sum of 2 Regular Hexahedral Dice
10
11
12
What is the expected value?
Outcome
Probability
Product
2
(1/36)
2(1/36)= 2/36
3
(2/36)
3(2/36)=6/36
4
(3/36)
2(3/36)=6/36
5
(4/36)
2(4/36)
6
(5/36)
2(5/36)
7
(6/36)
2(6/36)
8
(5/36)
2(5/36)
9
(4/36)
2(4/36)
10
(3/36)
2(3/36)
11
(2/36)
2(2/36)
12
(1/36)
12(1/36)
Expected value = sum of the products = (2+3+3+4+4+5+5+5+5+ . .. +12)/36 = 7
Common Core State
Standards
S.MD.2 Calculate the expected value of a
random variable; interpret it as the mean of the
probability distribution.
Fred’s Fun Factory
A famous arcade in a seaside resort town consists of many different
games of skill and chance. In order to play a popular “spinning wheel”
game at Fred’s Fun Factory Arcade, a player is required to pay a small,
fixed amount of 25 cents each time he/she wants to make the wheel spin.
When the wheel stops, the player is awarded tickets based on where the
wheel stops and these tickets are then redeemable for prizes at a
redemption center within the arcade.
The wheel awards the tickets with the following probabilities:
Number of Tickets
Probability
1 ticket
35%
2 tickets
20%
3 tickets
20%
5 tickets
10%
10 tickets
10%
25 tickets
4%
100 tickets
1%
a) If a player were to play this game many, many times, what is the expected
number of tickets that the player would win from each spin?
b) The arcade often provides quarters to its customers in $5.00 rolls. Every day
over the summer, Jack obtains one of these quarter rolls and uses all of the
quarters for the spinning wheel game. In the long run, what is the average
number of tickets that Jack can expect to win each day using this strategy?
c) One of the redemption center prizes that Jack is playing for costs 300 tickets. It
is also available at a store for $4.99. Without factoring in any enjoyment gained
from playing the game or from visiting the arcade, would you advise Jack to try
and obtain this item based on arcade ticket winnings or to buy the item from the
store? Explain.
Expected Value
Common Core State Standards
Use probability to evaluate outcomes of
decisions.
Weigh the possible outcomes of a decision by
assigning probabilities to payoff values and
finding expected values.
The Amore Ristorante Valentine Special
from Bock, D. E., Vellemna, P.F., DeVeaux, R. D. (2010). Stats: Modeling the world, Third edition. Boston: Addison Wesley
From a deck of 4 aces, pick the Ace of
Hearts, win a $20 discount.
If the card picked is the Ace of
Diamonds, you get a 2nd chance; pick
again for Ace of Hearts and a $10
discount.
Winnings={$20, $10, $0}
• P($20)= P(Ace of Hearts)= One fourth of the
time, the Valentine couple gets $20.
• P($10)=P(Ace of Diamonds followed by Ace of
Hearts)= One twelfth of the time, the
Valentine couple gets $10.
• P($0) = P(Black ace on 1st or 2nd draw) =
Two-thirds of the time, the Valentine couple
gets $0!
=
…so how much?
For 120 couples, the restaurant will on average
pay out $20 to 30 couples and $10 to 10
couples. That is $600 + $100 or $700 for 120
couples.
Expected Value of a Random Variable (X)
Outcome
Ace
Hearts
Ace Diamonds then Ace
Hearts
Ace Diamonds then Black Ace
Black Ace
X
20
10
0
0
P(X)
.25
(.25)(.3333…)
(.25)(.666666…)
0.5
The expected value is the mean of
E(X)= 20(.25)+10(.25)(.3333…)+0(.25)(.6666….) +
0(.5)
5. 83
μ = E(X)=
Box Model for Expected Value
Let’s say that the Restaurant has made
reservations for 100 couples. They can expect to
give 100(5.83) or $583 in discounts!
The problem can be modeled as the sum of 100
draws from a box of tickets:
100
|0 0 0 0 0 0 0 0 10 20 20 20|
With replacement
The expected value is the number of draws
times the average of the box.
Expected Value
Common Core State Standards
(S.MD.3) Develop a probability distribution
for a random variable defined for a sample
space in which theoretical probabilities can
be calculated; find the expected value.
For example, find the theoretical probability
distribution for the number of correct answers
obtained by guessing on all five questions of a
multiple-choice test where each question has four
choices, and find the expected grade under
various grading schemes.
What do you expect?
A student guesses on all 5
questions of a multiple 4 choice
test.
5 correct?
4 correct?
3 correct?
2 correct?
1 correct?
The big zero?
What do you expect? Does the probability help you decide?
Binomial Formula
If trials are independent, the probability is
same for each trial, and the number of trials
is decided in advance then the probability of
exactly k occurrences in n trials is
k(1-p)n-k
C
•p
n k
Probability Histogram
(from TI 84 Calculator)
Expected Value
Grade
Frequency for 1000
0
0.237(1000)=237
1
0.396(1000)=396
2
0.264(1000)=264
3
0.088(1000)= 88
4
0.015(1000)= 15
5
0.001(1000)= 1
How would you calculate the average grade?
(0(237)+1(396)+2(264)+ …)/1000
Expected Value = 0(0.237) + 1(0.396) + 2(0.264)+… ≈1.25
A game consists of throwing a dart at a target.
Assume the dart must hit the target and land
inside one of the squares. The player pays $24
to play the game. If the dart hits the shaded
region, the player wins $44, otherwise the
player receives nothing. What is the expected
value of the game?
9
Probability of shaded region:
16
7
Probability of un-shaded region:
16
Profit: $44 - $24 = $20
9
Solution: (20)
16
-
7
(24) = .75 cents
16
The odds of winning a raffle are
1:3. If the winning prize is $12,
then how much should a ticket
cost if the raffle is fair?
1
(12
4
The ticket
should cost
$3
– c) +
3
4
−c = 0
12 1
−3
− 𝑐+
𝑐 = −3
4 4
4
−1
−3
𝑐+
𝑐 = −3
4
4
−1𝑐 = −3
For Activities & Resources
http://teachingex
pectedvalue.yola
site.com/