probability?

Probability?

Deal Or No Deal

Briefcase Presenters

• Violette Espinoza, Rickard’s MS• Sonia Kimbrough, Forest Glen MS• Wendy Moskowitz, Taravella HS• Lisa O’John, Forest Glen MS

Goals of Presentation

- Students are given the opportunity to explore probability and mean in a fun and familiarreal-world context.

- Students will apply mathematical reasoning to everyday decision making.

Florida Sunshine State Standards

New Standards• MA.7.P.7.2• MA.7.A.3.2• MA.8.A.6.2

Let’s Play a Game!

For ExampleGame 1

Game 2

First Suitcase Second Suitcase Third Suitcase $10 $20 $30

First Suitcase Second Suitcase Third Suitcase $5 $10 $45

Game 3First Suitcase Second Suitcase Third Suitcase $0 $0 $60

Formula Intuitively

€

Game 110 + 20 + 30

3= 20

€

Game 25 +10 + 45

3= 20

€

Game 30 + 0 + 60

3= 20

€

Median

20

10

0

Which game is most skewed and why?

What is Expected Value?Expected Value (EV) represents how much money one can expect to win or lose in a game.

The Expected Value is also known as the mean.

A Mathematical Breakdown

Game 1

€

E(V ) = 13 ⎛ ⎝ ⎜

⎞ ⎠ ⎟10( ) + 1

3 ⎛ ⎝ ⎜

⎞ ⎠ ⎟ 20( ) + 1

3 ⎛ ⎝ ⎜

⎞ ⎠ ⎟ 30( )

€

Factor out 13

.

€

Therefore, we will get the following :13

10 + 20 + 30( ) = 20

A Mathematical Breakdown

Game 2

€

E(V ) = 13 ⎛ ⎝ ⎜

⎞ ⎠ ⎟ 5( ) + 1

3 ⎛ ⎝ ⎜

⎞ ⎠ ⎟10( ) + 1

3 ⎛ ⎝ ⎜

⎞ ⎠ ⎟ 45( )

€

Factor out 13

.

€

Therefore, we will get the following :13

5 +10 + 45( ) = 20

A Mathematical Breakdown

Game 3

€

E(V ) = 13 ⎛ ⎝ ⎜

⎞ ⎠ ⎟ 0( ) + 1

3 ⎛ ⎝ ⎜

⎞ ⎠ ⎟ 0( ) + 1

3 ⎛ ⎝ ⎜

⎞ ⎠ ⎟ 60( )

€

Therefore, we will get the following :23

0( ) + 13

60( ) = 20€

E(V ) = 23 ⎛ ⎝ ⎜

⎞ ⎠ ⎟ 0( ) + 1

3 ⎛ ⎝ ⎜

⎞ ⎠ ⎟ 60( )

How to apply formula to Deal or No Deal

• Deal or no Deal is a chance-based TV show.• A player is given a gallery of 26 closed, money-

filled briefcases.• Ranging in value from $.01 to $1,000,000. • The game is simple because there is no content

knowledge involved, but understanding EV can help a player decide what to do.

Formula for Expected Value of Deal or No Deal

€

E(X) = x i *P(x i)i=1

26

∑

Formula of EV

What do these symbols and words mean?∑ = sum

xi = represents the amount of money in each suitcase at that given moment.

P(xi) = the probability of the outcome happening.

€

EVevent = x i *P x i( )i=1

26

∑

Formula for Expected Value of Deal or No Deal

€

E(X) = x i *P(x i)i=1

26

∑

€

x1 → .01⇒ P(.01) = 126

€

.01* 126

+

€

x2 → .1⇒ P(.1) = 126

€

.1* 126

+...+

€

1,000,000* 126

− − − − − − − − − − −$131,478 = E(X)

How to apply formula to Deal or No Deal

This total represents the average or mean of what a player can win. This number is the way that contestants can determine if the banker’s offer is reasonable.

Worksheet for Students

• This worksheet can be used to calculate the Expected Value at any round of the game.

• An Excel worksheet can also be used.

Let’s Play a Game!

First Question?

Answer: The briefcase is still part of the EV, but the value of the briefcase is unknown.

• 1. What role does “your briefcase” play in Computing the EV in the game as a whole?

Second Question?

• 2. How do you compute the EV after each round in Deal or No Deal?

We take the sum of all the opened cases, and then subtract it from the total amount ( which is $3,418,416.00). We then divide the amount by the remaining unopened cases.

Answer:

Third Question?

• 3. How does the EV change throughout game play?

Answer:The expected value changes based upon the remaining values of unopened cases.

Fourth Question?

• 4. How do the Banker’s offer and the EV compare following each round?

Answer: The Banker’s offer is generally lower than the EV.The patterns that we analyzed was a 70% decrease of the EV within the first2 to 3 rounds. As we continued to play,

Fifth Question?

• 5. During which period does the Banker want you to deal or no deal?

Answer: The banker usually wants you to accept the deal early in the round.

References:1.) www.ithaca.edu/faculty/cduncan/311/pascal-long.doc

”The Expected Value Theory of Rational Choice”

2.) “Winning Big Money”, Mathematics Teaching in the Middle School, Vol.14, No.6, February 2009

SOME MATH JOKESTeacher: “Who can tell me what 7 times 6 is ?”

Q: How does a mathematician induce good behavior in her children?

Mathematicians never die—the only lose some of their functions!

Q: What does the little mermaid wear?A: An Algae-bra.

A. I’ve told you n times, I’ve told you n+1 times….

Teacher: ”Very good!- And who can tell me what 6 times 7 is?

Student: “It’s 42!”

Same Student: “It’s 24!!”

THE END

MA.7.P.7.2: Determine, compare and make predications based on experimental or theoretical probability of independent or dependent events.

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MA.7.A.3.2: Add, subtract, multiply and divide integers, fractions and terminating decimals and perform exponential operations with rational bases and whole number exponents including solving problems in everyday contexts.

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MA.8.A.6.2: Make reasonable approximations of square roots and mathematical expressions that include square roots and use them to estimate solutions to problems and to compare mathematical expressions involving real numbers and radical expressions.

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