<div class=Section1>PART I. Textbook problems from Moore & McCabe<![if !supportEmptyParas]> <![endif]>
Chapter 5.1: 5.4, 5.5, 5.9, 5.64, 5.68, 5.70
Chapter 5.2: 5.32, 5.36, 5.67
Chapter 6.1: 6.7, 6,15, 6.16, 6.22
Chapter 6.2: 6.33, 6.42, 6.43, 6.51, 6.81
(you may hand write this part, pen or pencil, doesn't matter)
<![if !supportEmptyParas]>PART II. An Essay question from Professor Lew.<span style="mso-spacerun: yes"></span>
<![if !supportEmptyParas]>(THIS PART MUST BE TYPED)<![if !supportEmptyParas]>
Maximum Length: One page, two if you are wordy (don't spend > hour on this OK?)
Spacing: Single or double-spaced.<span style="mso-spacerun: yes"> </span>
Guidance: This part is required for full credit
Assignment: Probability and Statistics were developed in large part by gamblers and many of the games are good examples of discrete probability problems. Let's examine the game of "roulette" more closely. Problems 5.31 and 6.83 in your textbook are directly related to the game of roulette, but here is a web site:
http://12.9.217.9/LasVegas/help/roulette.html
it will help you understand the game. While this is a class in statistics for Economics and Business Majors, if you understand conceptually what is happening at a roulette table or any other game of chance and if you can solve basic problems involving the mean and variance of random variables, you will gain some understanding of risk.
Assume the spins are independent and each one of the 38 numbers on the table have an equal chance (1/38) of having a ball land on it after the wheel is spun. If you want to see a representation of this try:
http://www.myrecroom.com/products/403_pic.html
One of the simplest bets in Las Vegas is to choose EVEN or ODD. An Even/Odd bet is called an "outside bet" in Las Vegas and so only the numbers 1-36 can be counted. If the ball lands on 0 or 00, no matter how you bet, you will lose. So for example, suppose you bet on ODD. If the ball lands on even number, 0, or 00 you lose. If the ball lands on one of the odd numbers, you win. The outcomes in Las Vegas are pretty much binary: you either win or lose.
The payout for your bet is "Even Money" which means, if you bet $5 and win, you get your $5 back and an additional $5. If you bet $5 and lose, you lose $5. Your outcomes from this game (either +5 or -5) can be thought of as a discrete random variable.
OK, you step up to the roulette table ready to make a series of $5 bets on either even or odd (but not both). You've got $250, so you can eventually make 50 bets. Before you bet, answer the following questions:
Question 1. Construct a table like this and fill it out: (you don't need to type it)
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Outcome (f(x)) |
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Probability (p(x)) |
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Question 2. What is the mean (expected value) of this random variable (winnings)? What is its variance and standard deviation?
Question 3. Theory tells us that the distribution of mean winnings will approximately be normal for large samples (large is usually understood to be greater than or equal to 15). Given that this is true, use normal calculations (that means Z) to figure out the range (in dollars) in which your mean winnings are usually expected to fall 50% of the time if you make the following number of bets:
15 bets (multiply by 15 to get the dollar amounts)
30 bets (multiply by 30 to get the dollar amounts)
50 bets (multiply by 50 to get the dollar amounts)
(You need to think what the "middle 50%" would be defined as if you are looking at a normal distribution).
Question 4. This is related to question 3. What is the probability that you will make money (that is, end up with winnings of greater than zero dollars) if you make:
15 bets
versus
30 bets
versus
50 bets
(your Z scores might be greater than what table A lists, if that happens, just report the Z and write that the probability is near zero)
Question 5. Now that you've learned a little bit about gambling, in theory, how do you think you would gamble if this were real? (This question is just for fun, but I would like to see your answers)
Final Instructions:<![if !supportEmptyParas]> <![endif]>:WHEN YOU HAND THIS IN, STAPLE PART I and PART II TOGETHER