M302, Spring 2005 (Smith)

 INFORMATION FOR THIRD EXAM

 
Date: Friday, May 6

Time: Usual class time (12 noon - 12:50 p.m.)

Place: WEL 1.308 (Same room as first two exams; see map on information for first exam.)

 
What you may and may not bring to the exam:

• You should bring a pencil, eraser and scratch paper.
•  You may (and probably should) bring a non-programmable calculator.
•   You may not bring notes, your textbook, or any programmable device such as a Palm Pilot, cell phone, or graphing calculator.

Exam procedures:

•       Please sit every other seat in the auditorium.
•      Cell phones and pagers should be turned off before the exam begins.

Exam coverage: The quiz will cover Sections 7.1 - 7.7, 8.1, and 8.2.

Exam format: The quiz will be "semi-take-home." By that I mean that the quiz questions will be very similar to some selection from the following list of questions. By "very similar" I mean, for example, that the numbers might be changed or the wording might be different, or the setting might be different but the same techniques can be used. (The retake at the final exam time will be a slightly different selection, but also taken from the questions below.) You are allowed to get help with these questions before the exam from anyone except me or the TA or any other math department faculty member or TA .  (You may ask me or the TA questions about anything assigned for class or covered in class, but not about the questions below or about very similar questions, unless they have been assigned in the reading, assigned as homework, or discussed in class.) As stated above, you may not bring any notes to the exam. Also, be prepared for the questions on the exam to be in a different order from the corresponding questions below. 

 Reminder: Unless told otherwise, you need to explain how you got your answers to receive full credit.

Posting grades: I will try to get exam grades posted on Blackboard by 10 p.m., Sunday, May 8.

Possible exam questions:

1. A student was asked to compute a certain probability. His answer was 2. Could this be correct? Explain.

2. You toss four dice. Find the probability that at least two of the dice will show the same number.

3. Suppose you have the 26 letters of the alphabet on separate cards in a hat. Suppose you pick out a card, write down the letter on it , put the card back in the hat, mix up the cards, pick out another card, and so on. Write an expression for the probability that on your first five draws your letters (in the order drawn) spell TEXAS.

4. A tetrahedral die is a die in the shape of a tetrahedron, with the numbers 1, 2, 3, 4 written on the four different sides. When you toss a tetrahedral die, it lands with one corner up and one side down, so you look at the number that lands down. Assume all sides are equally likely to land down.

1. What is the probability of rolling a 3 (that is, of having the 3 land down)?
2. Suppose you simultaneously toss two tetrahedral dice. Make an organized list or chart of all possible outcomes for the two numbers that land down. Label clearly.
3. When you toss two tetrahedral dice, what is the probability that both numbers landing down are the same? Explain.
4. When you toss two tetrahedral dice, what is the probability that the two numbers landing down have sum less than or equal to 3? Explain.

5. There are 8 balls in a container. The balls are labeled 1, 2, 3, 4, 5, 6, 7, 8. For each of the questions a, b, and c below, tell which of the expressions (i) - (vii) shown here answers the question, and explain why.

Here are the possible answers:

1. 8 x 8 x 8 x 8
2. (8 x 8 x 8 x 8)/(4 x 4 x 4 x 4)
3. (8 x 8 x 8 x 8)/(4 x 3 x 2 x 1)
4. (8 x 8 x 8 x 8)/(8 x 7 x 6 x 5)
5. 8 x 7 x 6 x 5
6. (8 x 7 x 6 x 5)/(4 x 4 x 4 x 4)
7. (8 x 7 x 6 x 5)/(4 x 3 x 2 x 1)
8. None of the above expressions.

1. You draw out one ball, write down its number, put it back in the container, shake up the container, draw out a second ball, write down its number, put it back in the container, and so on until you have drawn out 4 balls, putting each one back in the container and shaking up the container before you draw the next ball. How many possible lists of four numbers can you get? (For this part, lists with the same numbers but in different orders are considered different -- for example, getting the numbers 5, 2, 5, 4 would be a different outcome than getting the numbers 2, 5, 4, 5.)
2. You draw out one ball, write down its number, then draw out the second ball without replacing the first ball, write down its number, and so on, until you have four balls. How many different lists of numbers are possible? (For this part also, lists with the same numbers but in different orders are considered different -- for example, getting the numbers 2, 5, 7, 3 would be a different outcome than getting the numbers 7, 5, 2, 3.)
3. You draw out one ball, draw out another ball (without replacing the first one), and so on until you have four different balls. You then look at their numbers, not caring about which number was drawn first. How many different groups of numbers are possible? (Note that I've said "group" rather than "list," since order is not taken into account here -- so, for example, 2, 5, 7, 3 and 7, 5, 2, 3 would be considered the same groups.)

6. Write down an expression that tells you what the probability is that at least one person in a room of 50 has their birthday today. Explain how you got your expression. (Hint: There are two ways to do this; one is much shorter than the other.)

7. Suppose a University has no regulations about what courses a student can or cannot take. If it offers 200 courses, write an expression for the number of different possible combinations of four different courses a student could take. Explain how you got your expression.

8. A roulette wheel has 36 spaces numbered 1 to 36, with half of them red and half of them black. There are also two green spaces, labeled 0 and 00. There is little slot by each space, and a little ball that can fit into the slot. When the wheel is spun, the ball goes around and lands in one of the slots when the wheel stops. There are various kinds of bets that can be made.

1. What is the probability that the ball lands on 15? Explain.
2. What is the probability that the ball lands on a red space? Explain.
3. What is the probability that the ball lands on an odd number? Explain.
4. What is the probability that the ball lands on a number from 1 to 12? Explain.
5. If you bet $10 on odd, you get $20 back if the ball lands on an odd number.

1. What is the expected value of this bet? Show your steps.
2. What will the casino expect to make off of 100 bets like this?
3. What payoff would be needed to make this game fair?

1. What is the expected value of this bet? Show your steps.
2. What will the casino expect to make off of 100 bets like this?
3. What payoff would be needed to make this game fair?

9. Suppose you play a game consisting of someone handing you a dime and a quarter. You flip the dime and quarter and keep all the coins that come up heads. What is the expected value of that game? Show our work. [Hint: It should help to make an organized list of all possible outcomes and what you win or lose for each.]

10. You play the following game with a friend: You flip two coins. If they both come up heads, your friend gets a dollar. Otherwise, you keep the money he bet. How much should your friend bet so that the game is fair? Explain.

11. If you asked each student in a class how many pets they had ever had (in their entire life) and then made a graph (histogram) of the information you obtained, which of the following would it most resemble? Explain why.

If the histogram doesn't print, try this pdf file.

12. The criminal justice system wants to know about repeat offenders, so they gather 1000 ex-cons who were released two years ago and not arrested since then. They want to know how many ex-cons have committed felonies since their release. Of course, none of those present would want to answer such a question. So the authorities ask everyone in the room to secretly flip a coin. Then they ask people to raise their hands If they flipped a head OR committed a felony. 620 people raise their hands. Estimate how many of the 1000 ex-cons committed felonies since their release. Clearly show all work leading to your conclusion.

13. All athletes participating in a regional high school track and field championship must provide a urine sample for a drug test. Those who fail are eliminated from the meet and suspended from competition for the following year. Studies show that, at the laboratory that analyzes the urine samples, the drug tests are 95% accurate. That means that 95% of the time when the athlete has been using drugs, the test says "positive," and similarly, 95% of the time when the athlete has not been using drugs, the test says "negative." If only 40 out of the 1000 athletes at the meet actually use drugs, what fraction of the athletes who fail the test (i.e., get a positive test result) are falsely accused and therefore suspended even though they have not used drugs?

14. Discuss the validity of the reasoning in each of the following situations. Tell whether the reasoning is sound or not, and why.
    a. Sales of lemonade in a restaurant are positively correlated with ticket sales at the swimming pool a block away. Therefore increased use of the swimming pool causes an increase in the purchase of lemonade.  
    b. A report by the U.S. Commissioner of Narcotics on a study of 2,213 hardcore narcotic addicts in the Lexington, Kentucky, Federal Hospital showed that 70.4% smoked marijuana  before using heroin. This shows that marijuana use causes people to use heroin.