Math 231Q-01, Probability
Fall, 2000

 Review for Final

Time:  MWF 9:00 - MSB 303
Instructor: Professor Chuck Vinsonhaler
Office:  MSB 124
Phone:  486-3944
Web Page:
Office Hours:  MWF 10, and by appointment
Text: A First Course in Probability, Fifth Edition, by Sheldon Ross

Homework:  Homework will be assigned daily, and collected about once a week. Occasionally, I will give a quiz consisting of one or two homework problems instead of collecting homework. Homework and quizzes count 20% of your grade. You are encouraged to work together on homework. However, unless a problem is designated a group problem, you must turn in your own version of the solution. Answers must be explained in full sentences, with correct punctuation, spelling and format.

Exams:  There will be two exams and a final. Dates will be announced in class. Each exam counts 25% and the final 30%. You may use a calculator and one page (8 by 11) of notes during exams.

    Exam 1: Monday, October 2, 2000.   Review and Sample Exam
    Exam 2: Friday, November 17, 2000   Review and Sample Exam

Grading:  At the end of the semester your points will be totaled. Grades will be assigned on a fixed scale: A-90%; B-80%; C-70%; D-60%. Plusses and minuses are assigned in the standard way. For example, 67% is a C-.


  1. Combinatorial analysis – permutations, combinations, multinomial coefficients
  2. Axioms of probability – sample space, axioms, properties
  3. Conditional probability and independence – Bayes’ formula
  4. Random variables – densities, expectation, variance, Binomial, Poisson distributions
  5. Continuous random variables – Uniform, Normal, Exponential
  6. Jointly distributed random variables – joint distributions, sums of independent random variables.
  7. Properties of expectation – expectation and variance of sums, conditional expectation, moment generating functions
  8. Limit theorems – weak and strong law of large numbers, central limit theorem


1.  Due September 6:  p. 25 - 2,3,4,7,10,12,14,18,19,21,22,31 (may be resubmitted by 9/15)
2.  Due September 13: p. 54 - 3,5,7,9,16,18,20,23,27,28,56
3.  Due September 20: p. 104 - 1,2,4,5,9,13,15
4.  Due September 27: p. 106 - 18,19,20,24,27,34,36,50,53
5.  Due October 11: p. 173 - 14,19,20,21,23,25,37,38
6.  Due October 18: p. 179 - 40,42,49,51,55,58,66,70,73,74
7.  Due October 25: p. 232 - 1,5,6,7,8,11,12,14
8.  Due November 1: p. 232 - 15,16,17,18,21,23,27,30,31.  A coin is flipped 10 times. Calculate the probability of at most two heads. Then estimate this probability using the normal distribution.
9.  Due November 8: p. 293 - 2,3,6,8,9,11,12,20
10. Due November 15: p. 294 - 15,17,22,23,27
11. Due November 29: p. 293 - 33,34,35,41
12. Due December 6: p. 421 - 2,3,4,5,6,7,11,14,15. Also calculate the mgf for Uniform(0,1).

Extra Credit Problems: These are problems to be carefully written up and submitted before the last class of the semester for extra credit.

EC 1: p.19 - #32
EC 2: A 3-way argument breaks out in Probability about grades on an exam and the participants decide to settle the matter with a dual. They will draw straws to see who shoots first, second and third. Then they will stand on the vertices of an equilateral triangle and fire single shots in order, continuing the cycle until only one person is left standing. The shooter may aim wherever (s)he chooses. Alexi is known to be accurate 100% of the time, while Brenda is 75% accurate and Chuck 60% accurate. If Chuck draws the first shot he will miss intentionally because Alexi will surely fire at Brenda. Calculate the probabilities of survival for each of the contestants. There are 6 different cases and you will need to use geometric series.
EC 3: A fair coin is flipped repeatedly. Which sequence is more likely to appear first, HTHT or HTHH?  Which sequence is more likely to appear most often?


Math 231Q-01
Review for Exam 1
Fall, 2000

Basic Counting Principles
? Count the choices at each stage and multiply
? Count the choices for each case and add

Arranging (Permutations): Draw a picture  ___  ___  ___
Choosing (Combinations): Use binomial coefficients

Sample Spaces and Events
? Axioms
? Venn Diagrams
? Equally Likely Outcomes: P(E) =

Conditional Probability
? Bayes Theorem (calculate the probability in each case and add)
? Multiplication Rule  (calculate the probability at each stage and multiply)
? Independence

Random Variables
? Cumulative Distribution Function (cdf):    F(b)
? Probability Mass Function (pmf):    p(x)

Sample Exam 1 Problems.

1.  How many different letter arrangements can be made using (all) the letters in SCIENCE?
2.  In a certain club, 45% enjoy red wine, 35% enjoy white and 20% enjoy both. What percentage enjoy at least one of the two types of wine?
3.  Urn A contains 3 white and 1 black ball; urn B contains 3 white and 5 black; urn C contains 3 yellow and 1 white ball. An urn is chosen at random and a ball drawn.
    A.What is the probability the ball is white?
    B. Given the ball is white, what is the probability urn C was chosen?
    C. Let X be the random variable that is 0 if a white ball is drawn, 1 if a black ball is drawn and 2 if a yellow ball is drawn. Describe the cmf and pmf for X.
4.  There is a 50% chance that the Queen carries the gene for hemophilia. If she is a carrier then each prince (male child) has a 50% chance of having hemophilia. Suppose the Queen has 3 princes and none have the disease.
    A.  What is the probability the queen is a carrier?
    B.  If the fourth child is a prince, what is the probability he will have homophilia.
    C.  Let E be the event that the first 3 princes do not have hemophilia and F the event that the fourth has no hemophilia. Are E and F independent?

1. 1260
2. 60%
3A. 11/24
3B.  2/11
3C. The cmf for X has 3 "jumps." Where and what size are they?
4A. 1/9
4B. 1/18
4C. First, what do you think? We want to show that the probability of F given E is the same as the probability of F (if we can).

Here is a more detailed solution of Problem 4.
E: first three princes do not have hemophilia
F: fourth does not have hemophilia
Q: Queen carries the gene.

4A. P(Q given E) = P(E and Q)/P(E)
P(E) = P(E and Q) + P(E and Q complement) = (½)(½)(½)(½) + (½)1 = 9/16.
Therefore P(Q given E) = 1/9

4B. P(F given E) = P(F and E)/P(E)
P(F and E) = P(F and E and Q) + P(F and E and Q complement) = (½)(½)(½)(½)(½) + 1/2 = 17/32
Therefore P(F given E) = (17/32)/(9/16) = 17/18.  Thus, P(F complement, given E) = 1/18.

4C.  P(F) = P(F and Q) + P(F and Q complement) = (½)(½) + ½ = 3/4. Since P(F) is not equal to P(F given E), the events E and F are dependent. This should be inituitively obvious. If hemophilia has not shown up in the first three princes, it indicates the Queen is not a carrier so the fourth prince will not have hemophilia.

Review for Exam 2

Discrete distributions
    probability mass function, cumulative distribution function
    Expected Value, Variance
    Binomial (n,p)
    Poisson (lambda) (including approximation of Binomial, lambda = np)

Continuous Distributions
    probability density function, cumulative distribution function
    Expected Value, Variance
    Uniform(a,b), Normal(mu, sigma squared), Exponential(lambda)
    Normal approximation of Binomial, mu = np, sigma squared = np(1-p)
    Use of Normal(0,1) table

Joint Distributions
    F(x,y), p(x,y), f(x,y)
    Marginal distributions
    Sums of random variables

Suggested Review Problems: p. 293 - 2,3,9,20;      p. 232 - 1,6,11,15,18,23,30 ;
                                                p. 173 - 14,19,38,40,55,58,73

Sample Exam 2.

1.  Suppose on average there is one homicide a day in Manhattan. What is the probability that next week there will be fewer than 2 homicides in Manhattan?
2.  Suppose the number of hours people sleep in a night is normally distributed with mean 8 and standard deviation 1. One hundred people are selected at random. What is the approximate probability that more than 20 of these people will sleep more than 9 hours tonight?
3. If the continuous random variable X has pdf f(x) = cx(1-x), 0<x<1 and is 0 elsewhere, find E[X] and Var[X].
4.  Suppose X is a normal random variable with mean m and variance 25. Suppose that P(X>c)=0.1 and P(X>2c)=0.05. Find m and c.
5.  What is the approximate probability that if you toss a fair coin 200 times, the number of heads is strictly larger than 98 but smaller than 102?
6.  Suppose X and Y have a joint pdf given by f(x,y) = 3x, 0<y<x, 0<x<1.
    (a) Are X and Y independent?
    (b) Find E[Y].
    (c) Find P(X>1/2 given that Y<3/4).
    (d) Find P(X>2Y)
7.  Solve Buffon's Needle Problem, or some variant of it.

Answers to Sample Exam 2.  (My answers - check them!)

1.  8exp(-7)
2.  .1292
3.  1/2,1/20
4.  -4.595, 1.815
5.  .1664
6. (a) no
    (b) 3/8
    (c) under construction
    (d) 1/2
7.  See page 255 in the text

                                REVIEW FOR FINAL EXAM

Review Session: Monday, December 18, 3:30 PM, MSB 215

Key Concepts: Counting Principles, Equally likely outcomes, pmf, pdf, cdf, expected value, variance, conditional probability, independence, approximating binomial distribution with the normal, joint distributions, moment generating function, Markov inequality, Chebyshev inequality, Weak Law of Large Numbers, Central Limit Theorem.

1. Suppose X is a Poisson random variable with parameter 3 and Y is a Poisson random variable with parameter 5. If X and Y are independent, what is P(XY = 4)?

2. Suppose X1,…,X25 are independent and identically distributed random variables with mean 1 and standard deviation s. If  P(X1 + … + X25 >100) = 0.2, give an estimate of  s.

3. A drug is tried on a group of 100 people with AIDS. These people were found to live an average of 1 year longer than expected, with a standard deviation of 4 years. Suppose we let Xi represent the increase in expected lifetime of the ith person, and assume that the Xi are i.i.d with mean 0 and standard deviation 4. If Y is the average of the Xi for i = 1,2,…,100, what is the probability of the observed result, that is, Y > 1?

4. A person is dealt 3 cards from a standard 52 card deck. What is the probability that he will have exactly two different suits?

5. Two random variables X and Y have joint density f(x,y) = 8xy for 0< x<y and 0<y<1. Are X and Y independent? What is the probability X+Y < 1?

6. If the continuous random variable X has density f(x) = cx^(-5) for x > 1 and 0 otherwise, find E(X) and Var(X).

7. What is the approximate probability that if you roll a fair die 60 times you will get a 3 exactly 10 times? Use the normal approximation.

8. A fair die is rolled. If X is the number that comes up, what is the moment generating function for X?

9. Write a paragraph discussing the difference between the Weak Law of Large Numbers and the Central Limit Theorem.

Answers by C. Vinsonhaler

1. 605[exp(-8)]/4
2. s = 17.8
3. 0.0062
4. 234/425
5 (a)  No
   (b)  1/6
6.  E(X) = 4/3, Var(X) = 2/9
7.  0.14
8.  M(t) = [exp(t)+exp(2t)+exp(3t)+exp(4t)+exp(5t)+exp(6t)]/6
9.  See the text and class notes