Math 340                                          Spring 1995 
Exam 1  


Problem 1

An experiment consists of drawing 5 cards with replacement from 
a deck of cards.  Let S_{5} denote the
number of occurrences of either the number 3 or the number 4 (in five draws).

a. Give the range and the probability distribution of S_{5}. 
   (No need to simplify your final answer.)

b. What is the expected value of $S_{5}$.

c. Let A=(1.5, 4.25]. What is the probability that the number of 
   occurrences of either the number 3 or the number 4, in five draws, is 
   a number between 1.5 and 4.25?  Namely, calculate P(S_{5}\in A).  

Problem 2.

a. Prove:  (A \cup B) - A = B - (A \cap B).

b. Using the result of part (a) prove: P(A \cup B)+P(A \cap B) = P(A) + P(B).

Problem 3.

Let \Omega = \omega_{1}, \cdots, \omega_{10} be a sample space with 
10 sample points.  Let P be a probability measure on \Omega such that:

P(omega_{1}) = P(omega_{5}) = P(\omega_{10}) =\frac{1}{10} 
P(omega_{2})=P(omega_{4})=P(omega_{6})=P(omega_{8}) = frac{1}{20} 
P(omega_{3}) = \frac{1}{5}
P(omega_{7}) = 0 
P(omega_{9}) = \frac{3}{10}

Let X be a randon variable on \Omega such that:

X(omega_{1}) = X(omega_{8}) = X(omega_{7}) = 2.5 
X(omega_{2}) = X(omega_{3}) = 4 
X(omega_{4}) = X(omega_{9})= 5.25
X(omega_{5}) = 1
X(omega_{6}) = X(omega_{10}) = 8

a.Give the range and the probability distribution of X.

b.What is the expected value of X?

c.Let A = [2, 7.5].  What is the probability that P(X \in A)?  
  (No need to simplify your final answer.)

Problem 4.

a.You walk into a party wothout knowing anyone there.  
  There are 6 women and 4 men and you know
  there are 3 married couples.  In how many ways can you guess 
  who the couples are?  (Justify your answer.)

b.Three dice are rolled twice.  What is the probability that 
  they show the same numbers if the dice
  are {\em not} distinguishable?  (Justify your answer.)

c.Four shoes are taken at random from five different pairs.  
  What is the probability that there is at
  least one pair among them?  (Justify your answer.)