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Let
Ω consist of the set of positive integers. Consider the subsets
Describe in terms of
and their complements the following sets:
-
-
-
- The even integers greater than 12.
- The positive integers which
are multiples of six.
- The integers which are even and no greater than 6 or which are odd and
greater than 12.
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Let
Ω be the set of integers 0 through 10. Let
,
the odd integers in
Ω , and
the integers in
Ω which are even or
less than three. Describe the following sets by listing their elements.
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
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Consider fifteen-word messages in English. Let
the set of such messages
which contain the word “bank” and let
the set of messages which contain the
word “bank”
and the word “credit.” Which event has the greater probability? Why?
implies
.
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A group of five persons consists of two men and three women. They are selected
one-by-one in a random manner. Let
E
i be the event a man is selected on the
i th
selection. Write an expression for the event that both men have been selected by thethird selection.
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Two persons play a game consecutively until one of them is successful or
there are ten unsuccessful plays. Let
E
i be the event of a success on the
i th
play of the game. Let
be the respective events that player one, player two,
or neither wins. Write an expression for each of these events in terms of the events
E
i ,
.
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Suppose the game in
[link] could, in principle, be played an unlimited
number of times. Write an expression for the event
D that the game will be terminated
with a success in a finite number of times. Write an expression for the event
F that
the game will never terminate.
Let
and
for
.
Then
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Find the (classical) probability that among three random digits,
with each digit (0 through 9) being equally likely and each triple equally likely:
- All three are alike.
- No two are alike.
- The first digit
is 0.
- Exactly two are alike.
Each triple has probability
- Ten triples, all alike:
.
-
triples all
different:
.
- 100 triples with first one zero:
-
ways to pick two positions alike; 10 ways to pick the common value;
9 ways to pick the other.
.
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The classical probability model is based on the assumption of equally likely
outcomes. Some care must be shown in analysis to be certain that this assumption is good.A well known example is the following. Two coins are tossed. One of three outcomes
is observed: Let
ω
1 be the outcome both
are “heads,”
ω
2 the outcome that both are “tails,” and
ω
3 be the outcome
that they are different. Is it reasonable to suppose these three outcomes areequally likely? What probabilities would you assign?
.
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Source:
OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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