Chapter 5, Probability (About of the Exam)

Mat 212

Exam 2 Study Guide

3/14/07

Exam 2 (Wednesday 3/21/07) will cover Chapters 5 and 6. You should understand and be able to explain the following concepts, as well as furnishing appropriate examples of their use and applying them to solve problems.

Chapter 5, Probability (about ½ of the exam):

1. outcome, sample space, event
2. probability, probability distribution
3. probability of a certain event is 1, probability of an impossible event is 0 (warning – I disagree with your author! see p.131)
4. empirical versus theoretical probabilities, law of large numbers
5. expected value
6. mutually exclusive events
7. independent events
8. rules for probability with “and”, “or”, “not”
9. General computational methods (for counting outcomes in an event or probability of an event):

• tree diagrams
• organized lists or tables
• slots (Fundamental Counting Principle)
• combinations (nCr)
• permutations (nPr)

Chapter 5, Statistics (about 1/6 of the exam):

1. Measures of center (“typical” value): mean, median, mode
2. Measures of spread: range, variance, standard deviation
3. quartiles
4. Representing categorical data: pictograph, bar graph, frequency table, pie chart
5. Representing quantitative data: histogram, line graph, class frequency table, boxplot

Chapter 6, Fractions (about 1/3 of the exam):

1. fractions can be interpreted as: part of a whole, a ratio of two quantities, the result of a division
2. equivalent fractions
3. finding a common denominator
4. reducing fractions to “simplest form”
5. improper fractions
6. mixed numbers
7. modeling fraction operations with color tiles
8. fraction arithmetic: addition, subtraction, multiplication, division

Sample Exam Questions (for practice – not exhaustive!)

1. Suppose you roll two ordinary dice and add the results.

• What is the sample space?
• What is the most likely sum? What is the probability of that most likely sum? Explain.
• Find the probability that the sum is less than 5.

2. An electric clock is stopped by a power failure. What is the probability that the second hand is stopped between the 5 and 6?

3. You toss a coin 10 times and notice “heads” comes up more than 7 times. What is the probability of this happening if the coin is fair? Does this outcome (heads more than 7 times in 10 tosses) give strong evidence that the coin is not fair? Explain.

4. You toss a coin 4 times. Assuming P(H) = ½ on each toss, what is the probability of each of the following? Explain.

• HHHH
• HTTH
• exactly one head
• exactly two heads
• at least one head

5. Your math class is planning to choose a random group of three students to complain to the Dean about your instructor (hope not!). If there are 20 students in your math class, how many different ways can this group of 3 students be chosen?

6. License plate numbers in a certain state consist of 3 letters followed by 3 numbers. How many different license plate numbers are possible? If such a plate is chosen at random, what is the probability that the letters spell out “HOT”? That the three digits form an even number? That the letters spell out “HOT” and the three digits form an even number? That the letters spell out “HOT” or the three digits form an even number? That the letters do not spell out “HOT”?

7. Find the probability distribution for this experiment: guess randomly on a 3-question true/false quiz. What is the “expected” result? What does this mean?

8. You play a game involving the roll of a fair die. If you roll an even number, you pay twice the number of dollars as the number of “pips” on the die. If you roll an odd number, you win twice the number of “pips” on the die. What are your expected winnings each time you play the game once? What does this mean?

9. What does it mean for two events to be independent? Give an example of an experiment and two events in that experiment which are independent. Find two other events which are not independent.

10. Do the same as #9 for mutually exclusive.

11. How many tennis matches are needed in an elimination tournament with 16 players? In a round-robin tournament with 16 players? Explain.

12. The Math Department recently surveyed its majors regarding which elective courses they planned to take the next year. Of the 28 respondents, 12 wanted to study Geometry, 17 wanted to study Algebra, and 8 wanted to study Statistics. In addition, 9 wanted to study both Geometry and Algebra, 6 wanted to study both Algebra and Statistics, and 5 wanted to study both Geometry and Statistics. Finally, 5 wanted to study all three subjects.

• Draw a Venn diagram depicting the results of this survey.
• How many people wanted only Algebra?
• How many people wanted Algebra and Geometry but not Statistics?
• How many people wanted none of these three subjects?

13. The following are the amounts (rounded to the nearest dollar) paid by 10 students for textbooks during the fall term:

109162110155127142130557298

(a) Name three different ways of calculating the “typical” amount paid for textbooks by these students. Then, calculate each of these for the given data.

(b) Display the data with an appropriate chart or graph.

(c) Calculate the standard deviation for these data. What does the standard deviation measure?

14. Fifteen Americans were asked to pick their favorite fast food restaurant from among the following options: McDonalds (M), Wendys (W), Burger King (BK), Taco Bell (TB), Subway (S). The responses were as follows:

WTBSBKS

TBTBBKTBS

WMSTBM

(a) Is the variable “favorite fast food restaurant” categorical, or is it quantitative?

(b) Name two different styles of visual display which are appropriate for these data. Create the displays.

15. Suppose there are four numbers, and their mean is 20.

(a) Will the median also be 20? Explain.

(b) How small could the standard deviation be? Explain.

(c) How large could the standard deviation be? Explain.

(d) Find four numbers with mean = 20 and standard deviation = 2.

16. Name three different ways for interpreting fractions. Give an example of each.

17. What does it mean for two fractions to be equivalent?

18. Find a value for x so that the two fractions are equivalent: and

19. Use a drawing of 12 “color tiles” (squares) to model the following.

20. One-fourth of it is one-sixth. What is it?

21. How many halves are contained in three units? ______Model this question with an equation, and illustrate with an appropriate drawing.

22. Use a “color tiles” illustration to show why dividing by 1/3 is the same as multiplying by 3.

23. Five maids can clean seven houses in three days. How long would it take eight maids to clean 20 houses if all the work was done at the same rate?

24. Illustrate how many times 4/9 is contained in 6/9.

25. Perform the indicated operation.

26. Show “by hand” how to perform 3/15 + 2/25.

27. Reduce this fraction to simplest form: 2520/2700. Show your steps.