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Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:1
8.3: Sample Spaces and Events
EXPERIMENT An experiment is an activity
with observable results.
SAMPLE POINT : an outcome of an
experiment.
SAMPLE SPACE : the set consisting of all
possible sample points of an experiment.
EVENT : a subset of a sample space of an
experiment.
UNION OF TWO EVENTS
The union of the two events E and F is the
event E  F.
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Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:2
E  F  s  S : s  E or s  F .
INTERSECTION OF TWO EVENTS
The inter sec tion of the two events E and F
is the event E  F.
EF  s  S : s  E&s  F .
COMPLEMENT OF AN EVENT
The complement of an event E is the event
Ec.
E c  s  S : s  E  S  E.
MUTUALLY EXCLUSIVE EVENTS
E and F are mutually exclusive if E  F  .
DeMORGAN’S LAWS: E  F c  E c  F c
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Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:3
E  F c  E c  F c .
(True for any number of events)
In Exercise 1  6, let S  a, b, c, d, e, f be a
sample space of an experiment and let
E  a, b, F  a, d, f, and G  b, c, e be
events of this experiment.
1.Find the events E  F and E  F.
E  F  a, b, d, f;
E  F  a.
2.Find the events F  G and F  G.
F  G  a, b, c, d, e, f;
FG  
3.Find the events F c and F  G c .
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Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:4
F c  b, c, e;
E  G c  a, b  a, d, f  a
4.Find the events E c and F c  G.
E c  c, d, e, f;
F c  G  b, c, e  b, c, e
 b, c, e.
5.Are the events E and F mutually
exclusive?
Since E  F  a is not a null set, we
conclude that E and F are not mutually
exclusive.
6.Are the events E  F and E  F c mutually
exclusive?
E  F  a, b, d, f and
E  F c  a, b  b, c, e  b.
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Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:5
Since b is an element of both sets, they
are not mutually exclusive.
In Exercise 7  12, let S  1, 2, 3, 4, 5, 6,
E  2, 4, 6 F  1, 3, 5, and G  5, 6.
7.Find the event E  F  G.
E  F  G  2, 4, 6  1, 3, 5  5, 6
 1, 2, 3, 4, 5, 6
8.Find the event E  F  G.
EFG 
2, 4, 6  1, 3, 5  5, 6  .
9.Find the event E  F  G c .
E  F  G c 
1, 2, 3, 4, 5, 6 c  .
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Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:6
10.Find the event E  F  G c .
E  F  G c  1, 2, 3, 4, 5, 6.
11.Are the events E and F mutually
exclusive?
Yes, E  F  ;
that is, E and F, do not contain any
common elements.
12.Are the events F and G mutually
exclusive?
No. 5 is an element of both sets.
In Exercises 13  18, let S be any sample
space in E, F, and G be any three events
associated with the experiment.
Describe the given events using the
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Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:7
symbols , , and c .
13. The event that E and/or F occurs
EF
14. The event that both E and F occur
EF
15. The event that G does not occur
Gc
16. The event that E but not F occurs
E  F c 
17. The event that none of the events E, F,
and G occurs
E  F  G c
18. The events that E occurs but neither of
the events F or G occurs.
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Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:8
E  F c  G c 
19. Let S  a, b, c be a sample space of
an experiment with outcomes a, b, and c.
List all the events of this experiments.
, a, b, c, a, b,
a, c, b, c, a, b, c.
20. Let S  1, 2, 3 be a sample space
associated with an experiment.
a. List all events of this experiment.
a. , 1, 2, 3, 1, 2,
1, 3, 2, 3, 1, 2, 3
b. How many subsets of S contain the
number 3?
b. 4
c. How many subsets of S contain either
the number 2 or the number 3?
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Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:9
c. 4
2.391TAN22. An experiment consists of
selecting a letter at random from the letters
in the word MASSACHUSETTS and
observing the outcomes.
a. What is an appropriate sample space for
this experiment.
a. A, C, E, H, M, S, T, U
b. Describe the event "a head is tossed
and an even number is cost."
b. A, E, U
3.391TAN23. An experiment consists of
tossing a coin and casting a die and
observing the outcomes.
a. Describe an appropriate sample space
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Dr. Raja Latif. Math 131 (043) Sec8.3
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for this experiment.
a. S  H, 1, H, 2, H, 3, H, 4,
H, 5, H, 6, T, 1, T, 2, T, 3,
T, 4, T, 5, T, 6.
b. Describe the event "a head is tossed
and an even number is cast."
b. E  H, 2, H, 4, H, 6.
4. 391TAN25. Quality Control A sample of
three transistors taken from a local
electronics store was examined to
determine whether the transistors were
effective d or non-defective n.
What is an appropriate sample space for
this experiment?
S  d, d, d, d, d, n, d, n, d, n, d, d,
d, n, n, n, d, n, n, n, d, n, n, n
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Dr. Raja Latif. Math 131 (043) Sec8.3
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5.391TAN27. Game Shows In a television
game show, the winner is asked to select
three prizes from five different prizes,
A, B, C, D, and E.
a. Describe a sample space for possible
outcomes(order is not important).
a.ABC, ABD, ABE, ACD, ACE,
ADE, BCD, BCE, BDE, CDE.
b. How many points are there in the
sample space corresponding to a selection
that includes A.
b. 6
c. How many points are there in the
sample space corresponding to a selection
that include A and B.
c. 3
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d. How many points are there in the
sample space corresponding to a selection
that includes either A or B.
d. 6
6.392TAN35. Shutle Bus Usage A certain
airport hotel operates a shuttle bus service
between the hotel and the airport.
The maximum capacity of a bus is 20
passengers.
On alternate trips of the shuttle bus over a
period of 1 week, the hotel manager kept a
record of the number of passengers
arriving at the hotel in each bus.
a. What is an appropriate sample space for
this experiment?
a. S  0, 1, 2, . . . . . , 20
b. Describe the event E that a shuttle bus
carried fewer than ten passengers.
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Dr. Raja Latif. Math 131 (043) Sec8.3
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b. E  0, 1, 2, . . . , 9
c. Describe the event F that a shuttle bus
arrived with a full capacity.
c. F  20
7.393TAN38. Let S be a sample space for
an experiment and let E and F be events of
this experiment.
Show that the events E  F and E c  F c are
mutually exclusive.
Hint : Use De Morgan , s law.
E c  F c  E  F c
By De Morgans law.
Since E  F  E  F c  ,
they are mutually exclusive.
8.393TAN39. Let S be a sample space of
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Dr. Raja Latif. Math 131 (043) Sec8.3
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an experiment with n outcomes.
Determine the number of events of this
experiment.
The number of events of this experiment is
2n.
9.393TAN37. Let S be a sample space for
an experiment.
Show that if E is any event of an
experiment, then E and E c are mutually
exclusive.
If E is an event of an experiment then E c is
the event containing the element in S that
are not in E.
Therefore, E  E c   and the two sets are
mutually exclusive.
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Dr. Raja Latif. Math 131 (043) Sec8.3
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In Exercises 40  41 determine whether the
statements are true or false.
If it is true, explain why it is true.
If it is false, give an example to show why it
is false.
10.393TAN40. If E and F are mutually
exclusive and E and G are mutually
exclusive, than F and G are mutually
exclusive.
False. Let E  1, 2, 3,
F  4, 5, 6 ,and G  4, 5.
Then E  F   and E  G  ,
F  G  4, 5  .
11.393TAN41. The numbers 1, 2, and 3 are
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Dr. Raja Latif. Math 131 (043) Sec8.3
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written separately on three pieces of
paper. These slips of paper are then
placed in a bowl.
If you draw two slips from the bowl, one at
a time, without replacement, then the
sample space for this experiment consists
of six elements.
True. The sample space is
S  1, 2, 1, 3, 2, 1,
2, 3, 3, 1, 3, 2.
T8.3B9-14: Describe the nature of a
sample space for the given experiment,
and determine the nature of sample points.
10. Die Roll. Four dice are rolled, and the
number that turns up are observed.
Solution. 6. 6. 6. 6  6 4  1296.
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Dr. Raja Latif. Math 131 (043) Sec8.3
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12. Ball Selection. From an urn containing
eight different balls, four balls are drawn
successively without replacement.
Solution. 8. 7. 6. 5  1680.
14. Letter Selection. A four-letter "word" is
formed by successively choosing any four
letters from the alphabet with replacement.
Solution. 26. 26. 26. 26  26 4  456976.
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