section 4.3 rules of probabilitymayaj/m140_chapter4_sec... · probability 4 29 10 29 2 29 3 29 29...
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Section 4.3 Rules of Probability
Properties of Probability Distributions
1. 0 P (si) 1 for i = 1, 2, · · · , n.
2. P (s1) + P (s2) + · · ·+ P (sn) = 1
3. P ({si} [ {si}) = P (si) + P (si) (i 6= j) for i = 1, 2, · · · , n; j = 1, 2, · · · , n.
1. Let S = {s1, s2, s3, s4, s5, s6} be the sample space associated with an experiment having the
following partial probability distribution.
Outcomes s1 s2 s3 s4 s5 s6
Probability 429
1029
229
329
229
Consider the events: A = {s1, s2, s5}, B = {s3, s5, s6}, C = {s1, s3, s4, s6}, and D = {s1, s2, s3}
Calculate the following probabilities. (Give answers as fractions.)
(a) P (s2)
(b) P (D)
(c) P (Bc)
(d) P (Ac \ B)
2. The sample space S = {s1, s2, s3} has the property that P (s1)+P (s2) = 0.39 and P (s2)+P (s3) =
0.83. Find the probability of each outcome.
Definition: Two events E and F are said to be mutually exclusive if the two events have no
outcomes in common, that is E \ F = ?.
Properties of Probabilities: Revisited
1. 0 P (E) 1 for any event E.
2. P (S) = 1.
3. If E and F are mutually exclusive, then P (E [ F ) = P (E) + P (F ).
4. If E and F are any two events of an experiment, then P (E[F ) = P (E)+P (F )�P (E\F ).
(This should remind us of the union/addition rule from section 6.2.)
5. If E is an event of an experiment and Ec denotes the complement of E, then P (Ec) =
1�P (E). The reverse is also true, namely, P (E) = 1�P (Ec). (I call this the Complement
Rule)
3. An experiment consists of selecting a card at random from a 52-card deck. Refer to this experiment
and find the probability of the event
A diamond or a king is drawn.
4. Let E and F be two events of an experiment with sample space S. Suppose P (E) = 0.59,
P (F ) = 0.38, and P (Ec \ F ) = 0.28. Calculate the probabilities below.
(a) P (Ec)
(b) P (E \ F )
(c) P (Ec \ F c)
(d) P (E [ F c)
2 Spring 2018, Maya Johnson
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5. Let E and F be two mutually exclusive events, and suppose P (E) = 0.6 and P (F ) = 0.1.
Compute the probabilities below.
(a) P (E \ F )
(b) P (E [ F )
(c) P (Ec)
(d) P (Ec \ F c)
(e) P (Ec [ F c)
6. Among 500 freshmen pursuing a business degree at a university, 317 are enrolled in an economics
course, 214 are enrolled in a mathematics course, and 138 are enrolled in both an economics and a
mathematics course. What is the probability that a freshman selected at random from this group
is enrolled in each of the following? (Round answers to three decimal places.)
(a) an economics or a mathematics course
(b) exactly one of these two courses
(c) neither an economics course nor a mathematics course
3 Spring 2018, Maya Johnson
; a
÷57=500 ;NE7=3n ;nlM)=214 ;ntEnM=B8
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7. A poll was conducted among 250 residents of a certain city regarding tougher gun-control laws.
The results of the poll are shown in the table. (Round answers to three decimal places.)
Own Own Own a
Only a Only a Handgun Own
Handgun Rifle and a Rifle Neither Total
Favor Tougher Laws 0 11 0 139 150
Oppose Tougher Laws 64 4 21 0 89
No Opinion 0 0 0 11 11
Total 64 15 21 150 250
(a) If one of the participants is selected at random, what is the probability that he or she opposes
tougher gun-control laws?
(b) If one of the participants is selected at random, what is the probability that he or she owns
a handgun but not a rifle?
(c) If one of the participants is selected at random, what is the probability that he or she favors
tougher gun-control laws and does not own a handgun?
4 Spring 2018, Maya Johnson
O 0→ O oppose
0 0only handgun nls )
nls ) = 250
P( Oppose )=8zg1o= .3@
Pl handgun A rifle"
)= 62¥ = .2@
PL favors A handgun'
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