College Mathematics: Learning Worksheets Chapter 8
201
Name ________________________________ Date ______________ Class ____________
Goal: To find sample spaces for multiple experiments. To find events and probabilities for
the sample spaces
An experiment consists of drawing a single card from a standard 52-card deck.
In Problems 1–6, what is the probability of drawing
1. a red jack? 2. the ace of spades?
3. a 6? 4. a seven?
5. a five of hearts or a five of spades? 6. a card that is not a red six?
Section 8-1 Sample Spaces, Events, & Probability
Empirical Probability: frequency of occurance of ( )
() total number of trials
EfE
PE n
≈=
EnE
College Mathematics: Learning Worksheets Chapter 8
An experiment consists of dealing 4 cards from a standard 52-card deck. In Problems 7–12
what is the probability of being dealt a hand with
52! 270,725
7. 4 kings? 8. 3 kings and 1 queen?
4! 1
4! 4!
270,725
270,725
9. 2 kings and 2 queens? 10. 1 king and 3 queens?
4! 4!
4! 4!
270,725
270,725
11. no kings? 12. only face cards?
48! 194,580
12! 495
203
13. Sum is 8 14. Sum is 7
15. Sum is not 6. 16. Sum is divisible by 5.
7
(5 or 10) 36
P=
17. The sum is even. 18. Sum is greater than 1 and less than 13.
204
19. A Mathematics 110 class at a junior college must select 3 students to serve on an
advisory committee to the dean. There are 16 freshmen and 14 sophomores in the class.
a) i) How many 3-person committees can be chosen from the class?
30! 4060
ii) How many 3-person committees can be chosen from the class if the
committee consists of only freshmen?
14! 560
iii) How many 3-person committees can be chosen from the class if the
committee consists of only sophomores?
14! 364
iv) How many 3-person committees can be chosen from the class if the
committee consists of 2 freshmen and 1 sophomore?
16! 14!
b) What is the probability that the committee will consist of
i) all freshmen?
4060
ii) all sophomores?
4060
iii) 2 freshmen and 1 sophomore?
4060
iv) 1 freshman and 2 sophomores?
16! 14!
4060
v) Find the sum of the probabilities in parts (i)–(iv) and write an explanation
for the sum.
205
20. A computer store has 15 copies of a software package. Unknown to the store, 3 of the
packages have corrupted disks. What is the probability that
Denominator for parts c–f is
15,3
15! 455
3! 15 3 !
C
a) a customer buys one package, that it will be one of the packages that contains a
b) a customer buys one package, that it will be one of the packages that is not
c) a customer buys three packages and all are corrupted.
3,3
3! 1
3! 3 3 !
1
455
d) a customer buys three packages and two of them are corrupted.
3,2 12,1
2! 3 2 ! 1! 12 1 !
312 36
455
e) a customer buys three packages and one of them is corrupted.
3,1 12,2
1! 3 1 ! 2! 12 2 !
3 66 198
455
f) a customer buys three packages and none of them are corrupted.
3,0 12,3
0! 3 0 ! 3! 12 3 !
1 220 220
455
g) Find the sum of the probabilities in parts c–f and write an explanation for the sum.
College Mathematics: Learning Worksheets Chapter 8
206
College Mathematics: Learning Worksheets Chapter 8
207
Name ________________________________ Date ______________ Class ____________
Goal: To find probabilities of compound events. To find odds.
1. A container contains 9 red balls, 6 green balls, and 10 white balls. A single ball is drawn
at random from the container. Find the probability that
a) the ball is red. b) the ball is green or white.
25
25
c) the ball is red or green. d) the ball is not white.
25
25
e) the ball is not green. f) the ball is not red.
25
25
Section 8-2 Unions, Intersections, and
Complement of Events; Odds
Theorem: Probability of a Union of Two Events
For any two events: ()()()()PA B PA PB PA B
Complement: () 1 ( )PE PE or ( ) 1 ( )PE PE
Odds: In favor of event E: () ()
1() ()
PE PE
PE PE
() ()
PE PE
208
2. An experiment consists of rolling two fair dice and adding the dots on the two sides facing
up. Compute the probability of obtaining the following:
a) The sum is 5. b) The sum is 8.
c) The sum is not 5. d) The sum is not 8.
e) The sum is 7 or 11. f) The sum is 7 and 11.
62
36 36
82
36 9
=+
==
g) The sum is divisible by 3 or 4. h) The sum is divisible by 3 and 4.
209
3. A survey was taken at a local college campus. The survey asked if the student was a full-
time student or a part-time student and if he or she worked full time or part time. The
following chart illustrates the (empirical) results.
Did not work Worked part
time
Worked full
time
Total
Full-time student 0.15 0.22 0.14 0.51
If a student is picked at random, what is the probability that
a) he or she is a full-time student or works part time?
0.51 0.43 0.22 0.72
+− =
b) he or she is a full-time student and works part time?
c) he or she is a part-time student or does not work?
0.49 0.24 0.09 0.64
+−=
d) he or she is a part-time student and does not work?
e) he or she is a full-time student and does not work?
f) he or she is a part-time student or works full time?
PP P+−
211
5. A candidate for mayor of a small town predicts that he has an 75% chance of winning
the election.
a) what is the probability that he will lose the election?
100% 75%
25% 0.25
=−
==
b) what are the odds that he will win the election?
P
c) what are the odds that he will lose the election?
P
College Mathematics: Learning Worksheets Chapter 8
212
College Mathematics: Learning Worksheets Chapter 8
Name ________________________________ Date ______________ Class ____________
Goal: To find conditional probabilities using contingency tables
Given the probabilities in the table below for events in a sample space S, find the
probabilities in 1–9 relative to the probabilities in the table.
A B C D TOTALS
E 0.02 0.08 0.06 0.04 0.2
1. P (C) 2. P(F) 3. ()PA F
4. ( )PC E 5.
()PAF 6. ()PFA
0.05
( | ) 0.0625
0.8
PAF
==
0.05
(|) 0.714
0.07
PF A
=≈
7. ()PDE 8.
()PEB 9.
()PCC
0.2
0.18
0.36
Section 8-3 Conditional Probability, Intersection,
and Independence
Conditional Probability: ()
(|) ()
PA B
PAB PB
or ()
(|) ()
PB A
PB A PA
Product Rule: ( ) ()( | ) ()( | )PA B PAPB A PBPA B
214
10. Are events C and E independent? Why or why not?
11. Are events B and F independent? Why or why not?
12. Two balls are drawn from a container that holds 5 red balls, 7 green balls and
8 white balls.
a) If the first ball drawn is replaced before the second ball is drawn, find the
following probabilities:
i) that both balls are green.
77 49 0.1225
20 20 400
ii) that neither ball is green.
13 13 169 0.4225
20 20 400
iii) that at least one ball is green.