SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY
EXAM P
PROBABILITY
P SAMPLE EXAM SOLUTIONS
Copyright 2009 by the Society of Actuaries and the Casualty Actuarial Society
Some of the questions in this study note are taken from past SOA/CAS examinations.
PRINTED IN U.S.A.
1 of 66
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View Full Document1. Solution:
D
Let
event that a viewer watched gymnastics
event that a viewer watched baseball
event that a viewer watched soccer
G
B
S
=
=
=
Then we want to find
()
() () () ( ) ( ) ( ) ( )
Pr
1 Pr
1P
r
P
r
P
r
P
r
P
r
P
r
P
r
1
0.28 0.29 0.19 0.14 0.10 0.12 0.08
1 0.48 0.52
c
GBS
GBSG
BG
SB
SG
B
S
⎡⎤
∪∪
=−
⎣⎦
+
+
−
∩ −
∩ +
∩ ∩
=
−
++−−−+
=
−=

2.
Solution:
A
Let R = event of referral to a specialist
L = event of lab work
We want to find
P[R
∩
L] = P[R] + P[L] – P[R
∪
L] = P[R] + P[L] – 1 + P[~(R
∪
L)]
= P[R] + P[L] – 1 + P[~R
∩
~L] = 0.30 + 0.40 – 1 + 0.35 = 0.05 .

3.
Solution:
D
First note
[ ] [ ] [ ] [ ]
[]
[
]
[
]
''
'
PA B PA PB PA B
PA B
PA PB
∪=
+
−
∩
+
−
∩
Then add these two equations to get
[ ] [ ] [ ] [ ] [ ]
( ) [ ] [ ]
( )
(
)
[] []
'2
'
'
0.7 0.9
2
1
'
1.6
2
1
0.6
PA B PA B
PA
PB PB
P A B
A B
∪+
∪
=
+
+
−
∩+
∩
+=
+
−
∩
∪
∩
=+
−
=
2 of 66
4.
Solution:
A
()
() []
12
For
1, 2, let
event that a red ball is drawn form urn
event that a blue ball is drawn from urn
.
Then if
is the number of blue balls in urn 2,
0.44
Pr[
] Pr[
] Pr
i
i
i
Ri
Bi
x
R
RB
B
R
R
B
B
=
=
=
==
+
=
∩∪∩
∩
∩
[][] [][]
Pr
Pr
Pr
Pr
41
6
6
10
16
10
16
Therefore,
32
3
3
32
2.2
16
16
16
2.2
35.2 3
32
0.8
3.2
4
RR
BB
x
xx
x
x
x
+
⎛⎞
=+
⎜⎟
++
⎝⎠
+
=+=
+
+=
+
=
=

5.
Solution:
D
Let N(C) denote the number of policyholders in classification C .
Then
N(Young
∩
Female
∩
Single) = N(Young
∩
Female) – N(Young
∩
Female
∩
Married)
= N(Young) – N(Young
∩
Male) – [N(Young
∩
Married) – N(Young
∩
Married
∩
Male)] = 3000 – 1320 – (1400 – 600) = 880 .

6.
Solution:
B
Let
H = event that a death is due to heart disease
F = event that at least one parent suffered from heart disease
Then based on the medical records,
210 102
108
937
937
937 312
625
937
937
c
c
PH F
PF
−
⎡⎤
∩=
=
⎣⎦
−
and
108
108
625

0.173
937 937
625
c
c
c
PHF
∩
=
=
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