\input ignore.tex %This line causes the smce macros to be ignored.

\preface

\def\problemskip{\vskip0.5in}

\title
\noindent
{\bf Math 104}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam I}\hfill
Instructor:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\sl August 4, 1994}\hfill
Section:\hbox to 2.5truein{\hrulefill}

\bigskip\noindent
Calculators are not allowed.  Hand in this answer page only.  Record your answers to the
multiple choice problems by  placing an $\times$ through one letter for each problem on this
answer sheet. There are  20 multiple choice questions worth 5 points each.  You are taking
this exam under the honor code.

\bigskip\noindent
Signature:\hbox to 2truein{\hrulefill}

\bigskip\noindent
number wrong times 5 = \hbox to 1truein{\hrulefill}\hfill
Score \hbox to 1truein{\hrulefill}

\def\answerskip{\vskip0.25in}


\note %1
\problem
An experiment consists of tossing two dice and observing the numbers on the top faces.  Consider
the following events:

E = both numbers are greater than or equal to 3.

F = both numbers are less than or equal to 3.

G = the total of the two numbers is seven or greater

H = at least one of the numbers is a two.

Whioch of the following is a pair of mutually exclusive events?


\correct
E and G

\wrong
F and H

\wrong
G and H

\wrong
E and F

\wrong
F and G

\note %2
\problem
If a student is selected at random from a class, there is a probability of exactly $\frac 23$ that
the student likes the music of the Grateful Dead, of exactly $\frac 14$ that the student like Bach
organ music, and of exactly $\frac{3}{14}$ that the student likes both Bach organ music and the
music of the Grateful Dead.  Which of the follwoing possible class sizes is compatible with those
probabilities?

\correct
28

\wrong
56

\wrong
84

\wrong
70

\wrong
42

\note %3
\problem
A fair coin is tossed 10 times.  What is the probability of obtaining exacltly 3 heads?

\correct
$\frac{3}{10}$

\wrong
$\frac{15}{128}$

\wrong
$\frac{1}{8}$

\wrong
$\frac{5}{32}$

\wrong
$\frac{27}{256}$


\note %4
\problem
A gambler uses a biased coin for which the odds in favor of coming up heads when tossed are 4:3. 
What is the probability that the coin will show tails when tossed?

\correct
$\frac{3}{7}$

\wrong
$\frac{4}{7}$

\wrong
$\frac{3}{4}$

\wrong
$\frac{1}{4}$

\wrong
$\frac{1}{3}$


\note %5
\problem
A visiting foreign leader is to travel in a motorcade from an air base P to her hotel at Q.  For
security reasons, the motorcade is equally likely to take any of the routes from P to Q (traveling
always East or South).  Which of the five intersections A, B, C, D, or E is the motorcade most
likely to pass through?

\vbox{\vskip 2true in}

\correct
A

\wrong
B

\wrong
C

\wrong
C

\wrong
E


\note %6
\problem
Independent events E and F have probabilities Pr(E) = .6 and Pr(F) = .5.  What is Pr(E $\cup$ F)?

\correct
0.7

\wrong
0.9

\wrong
0.8

\wrong
0.6

\wrong
1


\note %7
\problem
Carla and Jose each have 2 nickels and a dime.  Carla gives one of her coins at random to Jose,
and then Jose gives one of his four coins, at random, to Carla.  Carla later finds out she
doesn't have enough change to buy a 25 cent bag of sweets.  What is the probability that Jose
gave Carla a dime?  


\correct
$\frac 13$

\wrong
$\frac 15$

\wrong
$\frac 14$

\wrong
$\frac 12$

\wrong
$\frac 16$


\note %8
\problem
A packet of mixed nuts contains equal numbers of peanuts, walnuts, pecans and almonds, and half as
many pistachios as peanuts.  A nut is picked from the packet at random.  What is the probability
the nut is a pistachio or almond? 

\correct
$\frac 27$

\wrong
$\frac 25$

\wrong
$\frac 14$

\wrong
$\frac 15$

\wrong
$\frac 13$


\note %9
\problem
Three friends play a game of catch with a ball.  Whoever has the ball is equally likely to throw
it to either of the other two players.  What is the probability that on the third throw the bll
will return to the player who threw it first.

\correct
$\frac 14$

\wrong
$\frac 13$

\wrong
$\frac 12$

\wrong
$\frac 15$

\wrong
$\frac 16$


\note %10
\problem
What is the probability that of four independently and randomly scheduled meetings in March at
least two will occur on the same day?

\correct
$\frac {P(31,3)}{31^4}$

\wrong
1 - $\frac {P(31,4)}{31^4}$

\wrong
$\frac {C(31,4)}{31^4}$

\wrong
1 - $\frac {C(31,4)}{31^4}$

\wrong
1 - $\frac {C(31,4)}{P(31,4)}$

\note %11
\problem
A pair of fair dice is rolled once.  What is the probability that both ofthe dice come up as 5
given that at least one of them came up as a 5?


\correct
$\frac 19$

\wrong
$\frac {1}{10}$

\wrong
$\frac {1}{11}$

\wrong
$\frac 18$

\wrong
$\frac 17$


\note %12
\problem
The table below gives the probability distribution of a random variable X.

\vbox{\vskip 2true in}
What is the variance of X?

\correct
$\frac 45$

\wrong
$\frac 15$

\wrong
$\frac 35$

\wrong
$\frac 25$

\wrong
1


\note %13
\problem
A fair coin is tossed at most four times, stopping after the first head is obtained.  What is the
expected value of the number of tosses? 

\correct
1 $\frac 78$

\wrong
1 $\frac 34$

\wrong
1 $\frac 38$

\wrong
1 $\frac 14$

\wrong
2


\note %14
\problem
An instrument in a weather satellite will funcion correctly provided at least one of two circuits
A, b is working.  Each circuit A or B contains 100 trasistors, which must all function if the
circuit is to work.  The probability that any particular one of the transistors will fail within
five years is 0.002.  Assuming that transistor failures are independent events, what is the
probability the instrument will function correctly for five years? 

\correct
$1 - (.002)^{200}$

\wrong
$(.998)^{200}$

\wrong
$1 - (1 - (.998)^{100})^2$

\wrong
$1 - (1 - (.998)^2)^{100}$

\wrong
$(1 - (.998)^{100})^2$


\note %15
\problem
Six cards are selected randomly without replacement from a standard deck of 52 playing cards. 
What is the probability that the cards consist of 3 cards of one denonimation and 3 cards of a
second denomination?

\correct
$\frac{P(13,2) C(4,3)^2}{C(52,6)}$

\wrong
$\frac{C(13,2) C(4,3)^2}{C(52,6)}$

\wrong
$\frac{P(13,2) P(4,3)^2}{P(52,6)}$

\wrong
$\frac{C(13,2) C(8,6)}{C(52,6)}$

\wrong
$\frac{C(4,2) C(13,3)^2}{C(52,6)}$


\note %16
\problem
The table belwo gives the prbability distribution table for an experiment in which a biased die is
rolled once and the number on the tip face recorded.

\vbox{\vskip 1true in}
Consider the following events.
\newline
E = \{1,2\}      F = \{1,2,4\}     G = \{1,3,5\}     H = \{6\}.
\newline
Which of the following is a pair of independent events.


\correct
G and H

\wrong
E and F

\wrong
E and H

\wrong
F and H

\wrong
E and G


\note %17
\problem
A bag contains 2 red balls and 2 green balls.  the balls are randomly drawn, one at a time,
without replacement, until no red balls are left in the bag.  Find the probability that exactly 3
balls will be drawn from the bag.

\correct
$\frac 23$

\wrong
$\frac 12$

\wrong
$\frac 13$

\wrong
$\frac 14$

\wrong
$\frac 16$


\note %18
\problem
A toy manufacturer makes dolls at factories at Boston, Houston and Los Angeles.  30\% of the dolls
are manufactured at Boston, and of those, 2\% are defective (their ears are too big).  40\% of the
dolls are made at Houston, and 3\% of those are defective.  Of the remaining dolls (made in L.A.),
4\% are defective.  What is the probability that a defective doll was made in Boston?

\correct
0.15

\wrong
0.25

\wrong
0.3

\wrong
0.2

\wrong
0.1


\note %19
\problem
A picnic hamper contains 4 apples, 3 oranges and 2 bananas.  What is the probability that if 4
pieces of fruit are randomly selected from the hamper, there will be at least one piece of each
type?

\correct
$\frac 49$

\wrong
$\frac 47$

\wrong
$\frac 58$

\wrong
$\frac{8}{35}$

\wrong
$\frac 35$


\note %20
\problem
Two people A, B play a dice game.  A single fair die is thrown, and one of them pays the other a
number of dollars equal to the number shown on the die.  If the number is 1, 3 or 6, A pays the
money to B, otherwise B pays the money to A.  The game is played 60 times.  What are the total
expected winnings for player A?

\correct
\$10

\wrong
\$8

\wrong
\$0

\wrong
- \$8

\wrong
- \$10







\end
