© Clark Creative Education Casino Royale

Dice, Playing Cards, Ideal Unit: Probability & Expected Value Time Range: 3-4 Days Supplies: Pencil & Paper

Topics of Focus:

- Expected Value

- Probability & Compound Probability

Driving Question “How does expected value influence carnival and casino games?” Culminating Experience Design your own game

Common Core Alignment:

o Understand that two events A and B are independent if the probability of A and B occurring S-CP.2 together is the product of their probabilities, and use this characterization to determine if they are independent. Construct and interpret two-way frequency tables of data when two categories are associated S-CP.4 with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. Calculate the expected value of a random variable; interpret it as the mean of the probability S-MD.2 distribution.

Develop a probability distribution for a random variable defined for a sample space in which S-MD.4 probabilities are assigned empirically; find the expected value.

Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding S-MD.5 expected values.

S-MD.5a Find the expected payoff for a game of chance.

S-MD.5b Evaluate and compare strategies on the basis of expected values.

Use probabilities to make fair decisions (e.g., drawing by lots, using a random number S-MD.6 generator).

Analyze decisions and strategies using probability concepts (e.g., product testing, medical S-MD.7 testing, pulling a hockey goalie at the end of a game).

Procedures:

A.) In “Expectations”, students encounter five different situations that involve expected value. The purpose of this is to build this skill prior to the project.

B.) In "The Inexpensive and Odd Game Room" (p. 8-12), students work with five different odd games that include a mix of conditional probability and regular probability. Students will then complete the reflection to analyze their results and make decisions.

C.) In "Casino Royale", students are tasked with being intelligent Game Makers and Game Players.

First, students will complete “Welcome to Casino Royale” (p. 13-14). This is a proposal for their own game. They will need to plan all aspects of the game and design all of the outcomes so that they make sense. All games will be played with “Swagg Cash” and each person will be given 125 Swagg Dollars. I suggest having students complete this activity and then the next day having the “Game Day”.

Students will need to create a sign advertising their game for the next day. The sign should briefly describe how the game is played. Each outcome should be defined with an assigned value. Students will need to bring everything they will need for their game the next day. They will need to be reminded!

D.) Suggested Logistics for Game Day. The students need to be split into two groups. “Game Makers” and “Game Players”. The Game Makers will host their game at a table or desk. The Game Players will go around to calculate expected values and play the games.

Print 2 pages of (p. 15) for each student. They will have one page to use as a Game Maker and one page to use as a Game Player.

All students will need a packet to record their Game Day participation (p 16-20). Make sure the Players are recording in the Player section and the Makers are recording in the Maker Section. The object of the game is to end with the most Swagg Cash.

I suggest setting a timer and having the groups switch roles Makers-Players at the halfway point or making the activity a two day event where each group plays their role for an entire period. Students will need 5-10 minutes at the end of the period to collect all of their Swagg and complete the reflection questions.

E.) After the games are finished, students will finish their reflections and determine who has the most overall swagg. They will add their Game Maker Swagg to their Game Player Swagg. The winner will be the Master of Swagg.

* Aspects of the project can be completed independently. The entire project does not need to be completed to have a great learning experience, though it is suggested because it will best scaffold the skills and context.

How do you make decisions? Some will opt for a random strategy, some will trust their gut and others will try to make decisions that make the most sense. One such mathematical strategy to analyze these types of decisions is expected value. Expected value is the most likely value that would occur if the situation happened infinitely many times. By understanding expected value, people can make stronger decisions and make numerical sense of a variety of problems.

In each of the following problems, create probability distribution tables, calculate the expected value and make a decision based on the data.

-Raffle Time-

Raffles are a popular way to raise money for different causes. Often a reward is given to attract people to buy the tickets. Two different raffles are going on. Which one gives a buyer the best chance to win? The Raffle Queen Snappy Raffle

1000 raffle tickets are sold for $3.00 each. There is 250 raffle tickets are sold for $1.00 each. There is one grand prize for $750 and two consolation prizes one grand prize for $150 and three consolation of $200 each that will be awarded. What is the prizes of $25 each that will be awarded. What is the expected value of one ticket? expected value of one ticket?

Outcome Outcome

Probability Probability

Which raffle has the higher expected value? If you are going to buy a ticket for one of the raffles, which one would you pick?

During the early 2000’s a strategy to defend NBA legend Shaquille O’Neal was to intentionally foul him as soon as he got the basketball. Teams felt that O’Neal, a historically bad foul shooter, would score less points if he took foul shots than if he took a regular shot in the game. This strategy became known as “Hack a Shaq”. In the early 2010’s a similar strategy has at times been used against star center Dwight Howard. Is this a good strategy? How does Howard’s expected values compare to O’Neal’s? Shaquille O’Neal Dwight Howard

In his career O’Neal made 58.2% of his field goal In his career Howard has made 57.7% of his field attempts during the game and 52.7% of his free goal attempts during the game and 57.6% of his throws. free throws.

Field Goal Attempts (taking a shot) Field Goal Attempts (taking a shot)

Outcome 2 0

Probability

Shooting 2 Free Throws Shooting 2 Free Throws

Outcome 2 1 0

Probability

Is “Hack-a-Shaq” an effective strategy for either O’Neal or Howard? Why or why not?

If a team debates whether they should intentionally foul Dwight Howard, what would you suggest?

On some standardized tests like the SAT, there is a penalty for getting a question wrong. On the SAT, a test taker earns 1 point for every correct answer and subtracts ¼ for each incorrect answer.

A test taker has a question blank and less than ten seconds to go in the test. The question has 5 multiple choice options, should the test taker guess or not?

Outcome 1 -1/4

Probability .2 .8

If the test taker can eliminate an answer choice, how much does it improve their odds?

Outcome 1 -1/4

Probability .25 .75

-Health Insurance Premiums-

In most businesses, companies are given health insurance premiums (how much it costs for each employee) from insurance companies based on the amount of usage from that business for the previous year. In this example, determine the appropriate insurance premium for this company.

An insurance agency has compiled data for a company into the following chart. How much should they charge as an annual premium for each employee in order to cover the costs?

Amount of Annual Claims Probability Last year the company paid a $525 monthly premium for their employees. They cannot afford any health insurance cost increases <4000 .35 and will need to pass along any additional costs to its employees. How much will the employees need to pay?

6000 .28

8000 .15

10000 .12

>12000 .10