M 362K, Spring 03, Smith

## Assignment for Friday, September 19:

#### To Hand In:

1. [No reasons required in this problem *only*.] Some of the equations
below say the same thing. Divide the equations into groups so that all the
equations in each group say the same thing, but equations in different groups
say different things. For each group, draw a Venn diagram which *best *says
the same thing as the equations in that group. (For example, if A and B are
mutually exclusive, then the circles that represent them should not overlap;
if A is empty, then there should not be a circle for A in the diagram.)

AUB = B
AUB = A
AB = A AB = B AB^{c}= A
AB^{c}= B A^{c}B=B A^{c}B=A
AB = Ø A = Ø B = Ø

*Reasons are required in the remaining problems!*

2. p. 16 #12

3. An instructor gave her class a set of 10 problems and the information
that the final exam would consist of a random selection of 5 of them. A certain
student has figured out how to do only 7 of the 10 problems.

a. How many different 5 problem exams could the instructor make
up? (Consider two exams to be the same if they have the same problems, even
if the order is different.)

b. How many of the 5 problem exams in part (a) will consist entirely of
problems that the student has figured out?

c. What is the probabilty that the exam will consist entirely of problems
the student has figured out?

d. How many of the 5 problem exams will consisst of 4 problems that the
student has figured out and one that they haven't?

e. What is the probability that the exam will consist of 4 problems that
the student has figured out and one that they haven't?

f. What is the probability that the exam will consist of at least 4 problems
that the student has figure out?

4. a. Joe got up bleary-eyed and late one morning. He reached without looking
into his sock drawer, simultaneously pulled out two socks (still without looking),
and put them on. The drawer had 7 socks, of which exactly 3 were red and
4 black.

i. How many possible pairs of socks could he have pulled out (assuming
that you can distinguish betweeen different red socks and can distinguish
between different black socks)?

ii. How many of the pairs in (i) consist of 2 red socks?

iii. What is the probability that Joe pulled out 2 red socks?

iv. How many of the pairs in (i) consist of 2 black sock?

v. What is the probability that Joe pulled out 2 black socks?

vi. What is the probability that Joe pulled out one red sock and one black
sock?

b. Suppose the situation is the same as above, except there are n socks
total in the drawer (still 3 red and the remainder black). What would n have
to be for the probability to be 1/2 that Joe pulled out 2 red socks?