Exam 2 Review

Covers Sections:  3.1 -- 3.4, 4.1

While every effort has been made to ensure the accuracy of the questions and solutions below, mistakes do sometimes occur.  If you discover an error please let me know, either in class, on the OpenLab, or by email to jreitz@citytech.cuny.edu.

1. During Restaurant Week in New York City, many local restaurants offer a complete meal at a fixed price (a prixe fixe menu).  An Italian Restaurant allows customers to choose between two appetizers, three different entrees, and two desserts for $19.95.  How many different meals are possible?
2. A student ID at a local university consists of 5 digits plus one letter.  

1. How many different IDs are possible?
2. How many IDs are possible if digits are not allowed to repeat?
3. How many IDs are possible if digits are not allowed to repeat, the first digit must be a 9, and the letter must be either A, B or C?

3. A multiple choice exam consists of 6 questions, each with four possible answers a, b, c or d.

1. How many different ways are there to complete the exam?
2. What is the probability of getting a perfect score? (NOTE: each problem has only one correct answer).

2. 
   Consider the experiment in which you roll two dice (each die has 6 sides).

1. How many outcomes are there?  List the sample space.
2. For each event below, list the outcomes that are included.  Then give the probability of the event.

1. event A: the first die is a 2.
2. event B: the first die is an even number and the second die is an odd number.
3. event C: at least one of the die is a 5.

3. 
   Consider the experiment in which you toss a coin three times in a row.

1. How many outcomes are there?  List the sample space.
2. For each event below, list the outcomes that are included.  Then give the probability of the event.

1. event A: the second coin is H (heads).
2. event B: there is at least one T (tails).

4.  
   A pet store completed a survey of customers regarding pet preferences and ownership.  The results for dog ownership and feelings about cats are displayed in the table.  Use them to answer the following questions.

3. What is the probability that a randomly selected person from the group

1. dislikes cats?
2. likes cats and does not own a dog?
3. dislikes cats or has no opinion about them?
4. dislikes cats or does not own a dog?
5. dislikes cats, given that they own a dog?
6. owns a dog, given that they dislike cats?

4. Are the events “dislikes cats” and “owns a dog” independent?  How do you know?
5. Are the events “no opinion about cats” and “owns a dog” mutually exclusive? How do you know?

1. If P(A)=0.3, P(B)=0.6, and P(A|B)=0.21, find the joint probability P(A and B).
2. The probability that a randomly selected student wears contacts is 0.58.  The probability that a student owns a pair of glasses given that they wear contacts is 0.95.  Find the probability that a student both wears contacts and owns a pair of glasses.
3. If P(A)=0.75, P(B)=0.52, and P(A and B)=0.31, find the probability P(A or B).
4. According to data collected at a certain intersection, 12% of all cars drive above the speed limit. In addition, 30% of all cars passing through the intersection are red. If 8% of all cars observed are both red and driving above the speed limit, what is the probability that a randomly selected car will either be red or will drive above the speed limit?

1. In how many ways can a committee of 3 people be selected from a group of 15 people?
2. A student will display four pieces of art side-by-side on the wall of a gallery.  In how many different ways can the four pieces be arranged?
3. A club with 35 members needs to choose four people to be president, vice president, treasurer, and secretary.  In how many ways can the jobs be assigned?
4. Five cards are selected at random from a standard deck of 52 cards.  What is the probability that the hand consists of 3 red cards and 2 black cards?
5. Three marbles are randomly chosen from a box contains 4 blue marbles, 7 red marbles, and 5 green marbles.  What is the probability that:

1. All three marbles are red?
2. One marble is blue and the other two are green?

7. 
   Does the following table represent a valid probability distribution? Why or why not?

8. 
   The 70 employees in a fast-growing startup company were surveyed to see how many months x each had been employed at the company.  The frequency distribution for x is given in the table.  Use the data to create a probability distribution for x, and find the mean and standard deviation.

9. 
   72% of New York restaurants are given an “A” grade by the city health inspectors.  Suppose two restaurants are selected at random.  Let x be the number (out of 2) of restaurants selected that received an “A” grade.  Create a probability distribution for x (HINT: use a tree diagram) and find the mean and standard deviation.

Answer Key

1. a.                  
   b.i.         
   b.ii.
   b.iii.         
   c.i.          
   c.ii.
2. a. There are 36 outcomes:  {1-1, 1-2, 1-3, 1-4, 1-5, 1-6, 2-1, 2-2, 2-3, 2-4, 2-5, 2-6, 3-1, 3-2, 3-3, 3-4, 3-5, 3-6, 4-1, 4-2, 4-3, 4-4, 4-5, 4-6, 5-1, 5-2, 5-3, 5-4, 5-5, 5-6, 6-1, 6-2, 6-3, 6-4, 6-5, 6-6}
   b.i.  A = {2-1, 2-2, 2-3, 2-4, 2-5, 2-6},  
   b.ii. B = {2-1, 2-3, 2-5, 4-1, 4-3, 4-5, 6-1, 6-3, 6-5},
   b.iii. C = {5-1, 5-2, 5-3, 5-4, 5-5, 5-6, 1-5, 2-5, 3-5, 4-5, 6-5},  
3. a. There are 8 outcomes:  {HHH, THH, HTH, TTH, HHT, THT, HTT, TTT}
   b.i.  A = {HHH, THH, HHT, THT},
   b.ii. B = {THH, HTH, TTH, HHT, THT, HTT, TTT},
4. a.i.
   a.ii.
   a.iii.
   a.iv.  
   a.v.
   a.vi.
   b. They are NOT independent (they are dependent), because is not equal to  
   c. Yes, they are mutually exclusive, because the two events cannot happen at the same time -- there are 0 people that both own a dog and have no opinion about cats.
5. a. Multiplication Rule:  
   b. Multiplication Rule:
   c. Addition Rule:
   d. Addition Rule:
6. a.
   b.
   c.
   d. 
   e.i.
   e.ii.
7. It is not a probability distribution, because we cannot have a negative probability (all probabilities must be between 0 and 1), and also because the sum of the probabilities does not equal 1.
8. Probability Distribution:

        Mean , Standard deviation

9. Probability Distribution:

        Mean , Standard deviation