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8
Chapter 2: Probability
2.1
A
= {FF},
B
= {MM},
C
= {MF, FM, MM}.
Then,
A
∩
B
= 0
/
,
B
∩
C
= {MM},
B
C
∩
=
{MF, FM},
B
A
∪
={FF,MM},
C
A
∪
=
S
,
C
B
∪
=
C
.
2.2
a.
A
∩
B
b.
B
A
∪
c.
B
A
∪
d.
)
(
)
(
B
A
B
A
∩
∪
∩
2.3
2.4
a.
b.
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Probability
9
Instructor’s Solutions Manual
2.5
a.
A
S
A
B
B
A
B
A
B
A
=
∩
=
∪
∩
=
∩
∪
∩
)
(
)
(
)
(.
b.
A
A
B
B
B
A
B
B
A
B
=
∩
=
∩
∪
∩
=
∩
∪
)
(
)
(
)
(
)
c.
=
∩
∩
=
∩
∩
∩
)
(
)
(
)
(
B
B
A
B
A
B
A
0
/
.
The result follows from part a.
d.
)
(
)
(
B
B
A
B
A
B
∩
∩
=
∩
∩
= 0
/
.
The result follows from part b.
2.6
A
= {(1,2), (2,2), (3,2), (4,2), (5,2), (6,2), (1,4), (2,4), (3,4), (4,4), (5,4), (6,4), (1,6), (2,6),
(3,6), (4,6), (5,6), (6,6)}
C
= {(2,2), (2,4), (2,6), (4,2), (4,4), (4,6), (6,2), (6,4), (6,6)}
A
∩
B
= {(2,2), (4,2), (6,2), (2,4), (4,4), (6,4), (2,6), (4,6), (6,6)}
B
A
∩
= {(1,2), (3,2), (5,2), (1,4), (3,4), (5,4), (1,6), (3,6), (5,6)}
B
A
∪
= everything but {(1,2), (1,4), (1,6), (3,2), (3,4), (3,6), (5,2), (5,4), (5,6)}
A
C
A
=
∩
2.7
A
= {two males} = {M
1
, M
2
), (M
1
,M
3
), (M
2
,M
3
)}
B
= {at least one female} = {(M
1
,W
1
), (M
2
,W
1
), (M
3
,W
1
), (M
1
,W
2
), (M
2
,W
2
), (M
3
,W
2
),
{W
1
,W
2
)}
B
= {no females} = A
S
B
A
=
∪
=
∩
B
A
0
/
A
B
A
=
∩
2.8
a.
36 + 6 = 42
b.
33
c.
18
2.9
S
= {A+, B+, AB+, O+, A, B, AB, O}
2.10
a.
S
= {A, B, AB, O}
b.
P
({A}) = 0.41,
P
({B}) = 0.10,
P
({AB}) = 0.04,
P
({O}) = 0.45.
c.
P
({A} or {B}) =
P
({A}) +
P
({B}) = 0.51, since the events are mutually exclusive.
2.11
a.
Since
P
(
S
) =
P
(E
1
) + … +
P
(E
5
) = 1, 1 = .15 + .15 + .40 + 3
P
(E
5
).
So,
P
(E
5
) = .10 and
P
(E
4
) = .20.
b.
Obviously,
P
(E
3
) +
P
(E
4
) +
P
(E
5
) = .6.
Thus, they are all equal to .2
2.12
a.
Let L = {left tern}, R = {right turn}, C = {continues straight}.
b.
P
(vehicle turns) =
P
(L) +
P
(R) = 1/3 + 1/3 = 2/3.
2.13
a.
Denote the events as very likely (VL), somewhat likely (SL), unlikely (U), other (O).
b.
Not equally likely:
P
(VL) = .24,
P
(SL) = .24,
P
(U) = .40,
P
(O) = .12.
c.
P
(at least SL) =
P
(SL) +
P
(VL) = .48.
2.14
a.
P
(needs glasses) = .44 + .14 = .48
b.
P
(needs glasses but doesn’t use them) = .14
c.
P
(uses glasses) = .44 + .02 = .46
2.15
a.
Since the events are M.E.,
P
(
S
) =
P
(E
1
) + … +
P
(E
4
) = 1.
So,
P
(E
2
) = 1 – .01 – .09 –
.81 = .09.
b.
P
(at least one hit) =
P
(E
1
) +
P
(E
2
) +
P
(E
3
) = .19.
10
Chapter 2:
Probability
Instructor’s Solutions Manual
2.16
a.
1/3
b.
1/3 + 1/15 = 6/15
c.
1/3 + 1/16 = 19/48
d.
49/240
2.17
Let B = bushing defect, SH = shaft defect.
a.
P
(B) = .06 + .02 = .08
b.
P
(B or SH) = .06 + .08 + .02 = .16
c.
P
(exactly one defect) = .06 + .08 = .14
d.
P
(neither defect) = 1 – P(B or SH) = 1 – .16 = .84
2.18
a.
S
= {HH, TH, HT, TT}
b.
if the coin is fair, all events have probability .25.
c.
A
=
{HT, TH},
B
= {HT, TH, HH}
d.
P
(
A
) = .5,
P
(
B
) = .75,
P
(
B
A
∩
) =
P
(
A
) = .5,
P
(
B
A
∪
) =
P
(
B
) = .75,
P
(
B
A
∪
) = 1.
2.19
a.
(V
1
, V
1
), (V
1
, V
2
), (V
1
, V
3
), (V
2
, V
1
), (V
2
, V
2
), (V
2
, V
3
), (V
3
, V
1
), (V
3
, V
2
), (V
3
, V
3
)
b.
if equally likely, all have probability of 1/9.
c.
A
= {same vendor gets both} = {(V
1
, V
1
), (V
2
, V
2
), (V
3
, V
3
)}
B
= {at least one V2} = {(V
1
, V
2
), (V
2
, V
1
), (V
2
, V
2
), (V
2
, V
3
), (V
3
, V
2
)}
So,
P
(
A
) = 1/3,
P
(
B
) = 5/9,
P
(
B
A
∪
) = 7/9,
P
(
B
A
∩
) = 1/9.
2.20
a.
P
(
G
) =
P
(
D
1
) =
P
(
D
2
) = 1/3.
b.
i. The probability of selecting the good prize is 1/3.
ii. She will get the other dud.
iii. She will get the good prize.
iv. Her probability of winning is now 2/3.
v. The best strategy is to switch.
2.21
P
(
A
) =
P
(
)
(
)
(
B
A
B
A
∩
∪
∩
) =
P
)
(
B
A
∩
+
P
)
(
B
A
∩
since these are M.E. by Ex. 2.5.
2.22
P
(
A
) =
P
(
)
(
B
A
B
∩
∪
) =
P
(B) +
P
)
(
B
A
∩
since these are M.E. by Ex. 2.5.
2.23
All elements in
B
are in
A
, so that when
B
occurs,
A
must also occur.
However, it is
possible for
A
to occur and
B
not
to occur.
2.24
From the relation in Ex. 2.22,
P
)
(
B
A
∩
≥
0, so
P
(
B
)
≤
P
(
A
).
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This note was uploaded on 10/24/2010 for the course BIOSTATIST 6057 taught by Professor Yu during the Fall '10 term at University of South Florida  Tampa.
 Fall '10
 yu

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