Math 211

Fourth Hour Exam

Name __________________________

Friday, Dec. 5

100 pts.

1. (15 pts.) If a fair (standard) die is thrown

What are the possible outcomes?

What is the probability that a 5 will be thrown?

What does this mean (in words)?

2. (5 pts. each)

a. Throwing one fair die, what is the probability of throwing an even number?

b. Define P(A|B)

c. Throwing one fair die, what is the probability that you throw a 3 given that you throw an odd number?

d. What is the probability of drawing a five-card hand containing exactly two aces?

e. Throwing two fair dice, what is the probability that the sum of the dice is even?

f. Suppose we throw a weighted die. The probability of throwing a 1 is three times the probability of throwing a 2, and the probabilities of throwing a 2, 3, 4, 5, or 6 are all equal (i.e., p(2) = p(3) = ... = p(6)). What are the probabilities?

g. A fair die is thrown quite a few times and the number that appears is recorded. These values are then averaged. What would you expect the average to be?

3. (15 pts.)

a. What is a Bernoulli trial?

b. What is the probability of throwing heads twice in five tosses of a fair coin?

c. Two fair dice are thrown. What is the expected number of times the two dice add up to 4 if the dice are thrown 10 times?

4. (5 pts.) Define what is meant (mathematically) by a relation between sets A and B

5. (10 pts.) A binary relation R is a relation in A: i.e., R is a relation between A and itself. What does it mean to say that the relation is

Reflexive?

Symmetric?

Anti-symmetric?

Transitive?

6. (5 pts. each)

a. What is an equivalence relation?

b. What does it mean to say that an equivalence relation partitions a set?

c. What is a partial order on a set A?

7. (5 pts.) Say something appropriate about one of the following:

a) Stephen Warshall

b) Papnuty Lvovich Chebyshev

c) James Bernoulli

d) Pierre-Simon Laplace

e) Paul Erdös