Download Math 132 Special Homework Assignment II

UNLV
University of Nevada, Las Vegas
The Department of Mathematical Sciences
Math 132 Special Homework Assignment II
Revised 6.18.2017
While all material covered in the text and listed on the syllabus is essential for success in the
course, the following material is of critical importance in Math 132. Problems included in this
document are a compilation of exam problems from previous semesters, and are similar to the
problems in the class text: Finite Mathematics, an Applied Approach, 10th edition.
1. In regard to the source of obtaining news, a survey of 1,600 people living in Las Vegas revealed that
650 used Newspapers (N), 550 relied on Radio (R), 1,100 viewed Television (T), 200 used both N &
R, 300 used R & T, 500 used T & N, and 200 used all the three sources N, R, and T. Using a Venn
diagram, answer the following questions
(a) How many used N or R?
(b) How many used N & T but not R?
(c) How many did not use T?
(d) How many did not use N?
2. How many ways are there to select a 6-person committee consisting of 3 students, 2 instructors, and
1 administrator from a group of 10 students, 5 instructors, and 3 administrators?
3. In how many ways can a nine-person committee, consisting of at most four Republicans and at most
seven Democrats, be chosen from a group of six Republicans and ten Democrats?
4. Using the appropriate formula and showing all your work, prove the following:
(a) C(n, k) = C(n – 1, k) + C(n – 1, k – 1)
(b) P(n, k) = P(n – 1, k) + k P(n – 1, k – 1)
5. How many different rearrangements of the letters in the word POSSESSION are possible if the letter
E must come before the letter P?
6. Find the coefficient of x 5 y 3 in the expansion of ( 3x - 13 y) 8
7. A fair coin is tossed seven times. Determine the following:
(a) How many possible outcomes are there?
(b) How many outcomes have (exactly) four “Heads”?
(c) How many outcomes have at most two “Tails”?
(d) How many outcomes have at least three “Tails”?
8. How many nonempty subsets, with even number of elements, are there of a set having eight
elements?
9. Combine the following sum of binomial coefficients into one single binomial coefficient.
 5  6  7  8
     
 5  5   5   5
10. Using a Venn diagram, indicate the set represented by ( A B C )
shading the area occupied by the set.
11. If c(A) = 25, c(B) = 15, and c(A
B) = 30, then find c(A
(A
B C) (A
B C ), by
B).
12. A pair of fair dice is rolled and the associated Sample Space S = (i, j ) :1  i  6,1  j  6 consists of
36 single outcomes represented by ordered pairs (i, j) in the sense that each time the experiment is
carried, one and exactly one of the 36 single outcomes occurs. Let E, F, G, H, and M be the events as
described below:
E = “the sum of the face values appearing is 7”
F = “the sum of the face values appearing is 11”
G = “at least one of the dice shows 1”
H = “one of the dice shows 3 or 4 and the other shows 2 or 3 or 4 or 5”
M = “the sum of the face values is 7 or at least one of the dice shows 1” = E G.
Q = “the sum of the face values is 7 and at least one of the dice shows 1” = E G.
V = “the sum of the face values is 7 and one of the dice shows 3 or 4 while the other shows 2 or 3 or
4 or 5” = E H
(a) Write each of the above events as a subset of the Sample Space S.
E=
F=
G=
H=
M=E G=
Q=E G=
V=E H=
(b) Answer the following giving your reason in each case:
(i) Are the events E and F mutually exclusive? Yes [ ], No [ ]. Reason:
(ii) Are the events E and G mutually exclusive? Yes [ ], No [ ]. Reason:
(iii) Is the event “E F” an “impossible” event? Yes [ ], No [ ]. Reason:
(c) Determine the following probabilities:
(i) P(E) =
(ii) P(F) =
(iii) P(G) =
(iv) P(H) =
(v) P(E G) =
Are the events E and G independent? Yes [
Are the events E and H independent? Yes [
(vi) P(E H) =
(vii) P(E/G) =
(viii) P(E/H) =
(ix) P(M) =
(x) P(N= M ) =
], No [ ]. Reason:
], No [ ]. Reason:
13. In a Casino game a pair of fair dice is rolled. If the event “M” as described in problem number 12
occurs you lose and the game is over. However, if the event “N = M ” occurs instead “M” a loaded
24
coin, with P(H) = 35
and P(T) = 11
35 on each single toss, is tossed. If the coin turns up “heads” you
win but if turns up “tails” you lose ending the game. With the help of a tree diagram, visualize a
Sample Space S for this game and determine the following:
(a) As a set S =
(b)
(c)
(d)
(e)
Probability of each single outcome listed in the Sample Space S
Probability that “you will win = W” P(W) =
Odds that “you will lose =L = W ”
Give an interpretation of the odds for losing.
(f) If you pay $2.00 to play the game each time and if you win, you win $2.75.What is the
expected value of winning in this game?
(g) Is the game fair? Yes [ ], No [ ]. Reason:
(h) How much should you win, if you win, to make the game fair?
14. In a family of 7 children, let E be the event that the family has at least 3 girls and F be the event that
the second born child is a girl. Find the following probabilities:
(a) P(E) =
(b) P(F) =
(c) P(E F) =
(d) P(E/F) =
15. A loaded coin, with P(H) = 53 and P(T) = 52 on each single toss, is tossed 6 times. Let E be the event
that (exactly) 3 heads appeared in 6 tosses and F be the event that at least 2 heads appeared in 6
tosses. Determine the following probabilities:
(a) P(E) =
(b) P(F) =
(c) P(E F) =
(d) P(E/F) =
16. The result of a recent survey of registered voters in Las Vegas in regard to their Gender (M=Male,
F=Female) and their preference of the political party (R=Republican, D=Democrat, I= Independent)
is given in the accompanying table. One of the voters was chosen at random by Channel 8 for an
interview for their news report. Determine the following:
a) P(R) =
R
D
I
Totals
b) P(R/M) =
M
750
150
600
1500
c) P(F) =
F 1750
1350 400
3500
d) P(F/D) =
Totals
2500
1500 1000
5000
e) P(M/D  I) =
f) Are the events R and M independent? Yes [ ], No [ ]. Reason:
g) Are the events F and D independent? Yes [ ], No [ ]. Reason:
17. In regard to the source of procuring news on current events, a recent survey of people living in Clark
County revealed that of 50% using News papers (N), 25% were Female (F) and 75% Male (M); of
20% using Radio stations (R), 40 % were F and 60% M; of 30% using Television channels (T), 65%
were F and 35% M. The survey also revealed that each person used only one of the three sources N,
R, or T.A person was chosen at random by Channel 8 for analysis. Determine the following in regard
to the selection of this individual:
(a) P(F) =
(b) P(R/F) =
(c) Based on this survey data, are the events R and F independent? Yes [ ], No [ ]. Reason:
(d) Are the events F and T independent? Yes [ ], No [ ]. Reason:
18. In a group of 5 people find the probability that at least 2 were born
(a) on the same day regardless of the week or month or year
(b) in the same month regardless of the date and year
(c) on the same date in the same month regardless of the year of 365 days
19. From a regular deck of 52 cards, 7 cards are drawn at random. Find the probability that the 7-card
hand drawn has
(a) exactly 3 Kings (Do Not Simplify the answer)
(b) exactly 2 Kings, 3 Queens, and 2 other cards (Do Not Simplify)
(c) exactly 3 Kings, 2 Queens, and a pair (Do Not Simplify)
20. In a raffle 1,500 tickets are sold at $2.75 each. The first prize is $1,500, the second prize is $1,000,
and there are 25 consolation prizes of $35 each. Determine the following:
(a) the expected value of a ticket
(b) the expected loss if you bought 120 tickets
21. During the last 10 year period, 150 faculties were hired at a university. The probability that a
Female was hired during this period is 40%. Find the following:
(a) Expected number of Male Faculty hired during this period.
Do Not Simplify answers in parts (b) and (c)
(b) Probability that at least one of 150 faculties hired was a Female.
(c) Probability that exactly 25 of 150 faculties hired were Male.
22. A loaded coin, with P(H) = 137 and P(T) = 136 on each single toss, is tossed 520 times. Find the
expected number of tails (T) appearing.
23. Given that P(E) = 0.3, P(F) = 0.6, and P(E  F) = 0.72, determine the following:
(a) P(E  F) =
(b) P(E  F ) =
(c) P( E  F ) =
(d) P(E/F) =
(e) Are the events E and F mutually exclusive? Yes [ ], No [ ]. Reason:
(f) Are the events E and F independent? Yes [ ], No [ ]. Reason:
24. Given P(A  B), P(B) and that the events A and B are mutually exclusive, find the odds for A.
25. Given P(A  B), P(B) and that the events A and B are independent, find the odds for A.
26. If the odds for the event F are a to b then show that P( F ) = b / (a + b)
Mathematics 132 Special Homework Assignment – II /Answers
1.
2.
3.
4.
5.
6.
7.
8.
(a) 1,000. (b) 300. (c) 500. (d) 950.
3,600.
9,780
(a) 2,730. (b) 792.
37,800.
-504
(a) 128. (b) 35. (c) 29. (d) 99.
127
9
9.  
6
10. Shade the appropriate outer parts.
11. 10.
11
12. (c) (i) 16 .(ii) 181 .(iii) 36
.(iv) 13 .(v) 181 .(vi) 181 .(vii) 112 .(viii) 16 .(ix) 125 .( x) 127 .
13. (c) 52 .(d )3to2.(h)$3.
99
57
14. (a) 128
.(b) 12 .(c) 128
.(d ) 57
64 .
864
2997
864
32
15. (a) 3125 .(b) 3125 .(c) 3125 .(d ) 111
.
16. (a)0.5.(b)0.5.(c)0.7.(d )0.9.(e)0.3.
17. (a)0.4.(b)0.2.
360
7920
55
18. (a)1  2401
 0.85.(b)1  20736
 1  144
 0.61805. (c)1  17267274024
17748900625  0.0271.
C (48,4)
(4,3).C (44,2)
C (11,1).C (4,2)
19. (a) C (4,3).
.(b) C (4,2).CC (52,7)
.(c) C (4,3).C (4,2).
C (52,7)
C (52,7)
20. (a) $2.25. (b) $60.
150 
25
125
21. (a) 90. (b) 1  (0.6)150 .(b) 
 (0.6) (0.4)
 25 
22. 240.
23. (a) 0.18. (b) 0.12. (c) 0.28. (d) 0.3.
24. 9 to 16
25. 0.55.