# 3.4: Chapter 3 Review

**At Grade**Created by: Bruce DeWItt

The expected value gives us the average result over the long term. We use expected value tables and the simple formula \begin{align*}EV=\left ( {Value 1}\right )\left ( {Prob 1} \right )+\left ( {Value 2} \right )\left ( {Prob 2} \right )+...\end{align*} to calculate the expected value. We can put everything together for a full probability analysis of a situation by using our probability calculations and other tools like a tree diagram. Casinos are cognizant of what the expected value is on any of their games and are confident, despite having to occasionally give away some substantial prizes, that their games will make them money in the long run. We cannot ever predict with certainty what is going to happen in a given situation, but we can always run a simulation to approximate what can happen. We will often use a random number generator or a table of random digits to help us run a simulation.

### Chapter 3 Review Exercises

1) Ten red marbles and 15 blue marbles are in a bag. A game is played by first paying $5 and then picking two marbles out of the bag without replacement. If both marbles are red you are paid $10. If both marbles are blue, you are paid $5. If the marbles don't match, you are paid nothing. Analyze this game and determine whether or not it is to your advantage to play.

2) When two dice are rolled, you can get a total of anything between 2 and 12.

a) Use the table of random digits in Appendix A, Part 1 to simulate rolling two dice 36 times. Begin on line 119. Make a chart displaying the different results that you get and how many times you get each result.

b) How close was your simulation to the theoretical probability of what should happen in 36 rolls?

3) A bag contains a $100 bill and two $20 bills. A person plays a game in which a coin is flipped one time. If it is heads, then the player gets to pick two bills out of the bag. If it is tails, the player only gets to pick one bill out of the bag. How much should this game cost to play if it is to be a fair game?

4) Suppose there are 38 kids in your Statistics and Probability class. Devise a system using a random digit table so that the teacher can randomly select 4 students to each do a problem on the board. Use line 137 from the random digit table to carry out your simulation and state the numbers of the four students who are selected.

5) A spinner with three equally sized spaces on it are labeled 1, 2, and 3. A bag contains a $1 bill, a $5 bill, and a $10 bill. A player gets paid the amount they pull out of the bag times the number that they spin. What should this game cost in order to be a fair game?

6) The table below shows the probabilities for how kids get to school in the morning.

Method | Bus | Walk | Car | Other |

Probability | 0.31 | 0.14 | 0.39 | ??? |

a) What must the

Othercategory have as a probability?

b) Describe how you would assign digits from a random digit table to set up a simulation for selecting a student to find out how they got to school.

c) Carry out your simulation for a total of 10 students and record your results. Use line 104 from the random digit table.

7) In an archery competition, competitors shoot at a total of 20 targets. The table below shows the probabilities associated with hitting the center of certain numbers of targets. Some shooters are perfect and hit the center of all 20 targets and the poorest shooters still hit the center of 15 targets.

a) What is the most likely number of centers that a shooter will hit?

b) What is the expected number of centers that a shooter will hit?

# of Centers | 15 | 16 | 17 | 18 | 19 | 20 |

Probability | 0.04 | 0.12 | 0.35 | 0.28 | 0.18 | 0.03 |

8) In a game of chance, players pick one card from a well-shuffled deck of 52 cards. If the card is red, they get paid $2. If the card is a spade they get paid $3. If the card is a face card, they get paid $5 and if the card is an ace they get paid $10. A player gets paid for all the categories they meet. For example, the King of Spades would be worth $8 because it is a spade and a face card. How much should this game cost in order to be a fair game?

#### Image References

Slot Machine http://www.gamedev.net

Quarter http://www.marshu.com

$10 Bill http://wingedliberation.tumblr.com

Welcome to Las Vegas http://pilipon.wordpress.com

Race Cars http://www.thunderboltgames.com

Electronic Devices http://www.topnews.in

Poker Chips http://www.ppppoker.com

Minnesota State Lottery http://www.mnlottery.com

$100 Bills http://www.sciencebuzz.org

MN Twins Logo www.twins.mlb.com

Final Exams Yes http://www.york.org

Pair of Jacks http://xdeal.com

Dice http://goblin-stock.deviantart.com

Targets http://www.theasbc.org

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