ACT245S
PROBLEM SET 4
1. Let (, ) , * and + be events such that
7 [*O(] ~ 12 , 7 [*O) ] ~ 34 ,
5
A) 32
B) 41
) ~ ( , * q + ~ J, and 7 [(] ~ 14 , 7 [) ] ~ 34 ,
7 [+O(] ~ 14 , 7 [+O) ] ~ 18 . Calculate 7 [* r +].
27
C) 32
Z
D) 43
E) 1
2. Let (Á ) and * be independent events such that 7 ´(µ ~ À Á 7 ´)µ ~ À
and
7 ´*µ ~ À . Calculate 7 ´( r ) r *µ .
A) À
B) À
C) À
D) À
E) À
Z
Z
3. A system has two components placed in series so that the system fails if either of the
two components fails. The second component is twice as likely to fail as the first. If the
two components operate independently, and if the probability that the entire system fails
is .28, find the probability that the first component fails.
A) À
B) À
C) À
D) À
E) lÀ
4. A ball is drawn at random from a box containing 10 balls numbered sequentially from
1 to 10. Let ? be the number of the ball selected, let 9 be the event that ? is an even
number, let : be the event that ? ‚ , and let ; be the event that ?  . Which of the
pairs ²9Á :³ Á ²9Á ; ³ Á and ²:Á ; ³ are independent?
A) ²9Á :³ only
B) ²9Á ; ³ only
C) ²:Á ; ³ only
D) ²9Á :³ and ²9Á ; ³ only
E) ²9Á :³ Á ²9Á ; ³ and ²:Á ; ³
5. If , Á , and , are events such that 7 ´, O, µ ~ 7 ´, O, µ ~ 7 ´, O, µ ~ Á
7 ´, q , µ ~ 7 ´, q , µ ~ 7 ´, q , µ ~ Á and 7 ´, q , q , µ ~ Á
find the probability that at least one of the three events occurs.
A) c B) c b
C) c b
D) c b
E) cb
6. ( writes to ) and does not receive an answer. Assuming that one letter in is lost in
the mail, find the chance that ) received the letter. It is to be assumed that ) would have
answered the letter if he had received it.
c
A) c
B) c
C) c
D) c
E) c
b 7. In conducting a survey it is found that all members of a population either read a
newspaper daily or have a driver's license, and those two attributes are independent of one
another. It is also found that for a person chosen at random from the population, there is
a probability of (   ) that the person has a driver's license. Find the probability
that a person chosen at random from the population reads a newspaper daily.
A. c B. c C. c D. b E. 8. A study of drinking and driving has found that 40% of all fatal auto accidents are
attributed to drunk drivers, 1% of all auto accidents are fatal, and drunk drivers are
responsible for 20% of all accidents. Find the percentage of non-fatal accidents caused by
drivers who are not drunk.
A. .780
B. .791
C. .802
D. .813
E. .824
ACT245S PROBLEM SET 4 SOLUTIONS
* and + have empty intersection, 7 [* r +] ~ 7 [* ] b 7 [+]À Also, since ( and )
are "exhaustive" events (since they are complementary events, their union is the entire sample
space, with a combined probability of 7 [( r ) ] ~ 7 [(] b 7 [) ] ~ 1). We use the rule
7 [* ] ~ 7 [* q (] b 7 [* q ( ] , and the rule 7 [*O(] ~ 7 [7([(q ]* ] to get
7 [* ] ~ 7 [*O(] h 7 [(] b 7 [*O( ] h 7 [( ] ~ 12 h 12 b 34 h 34 ~ 11
16 and
1
1
1
3
5À
7 [+] ~ 7 [+O(] h 7 [(] b 7 [+O( ] h 7 [( ] ~ 4 h 4 b 8 h 4 ~ 32
Then, 7 [* r +] ~ 7 [* ] b 7 [+] ~ 27
Answer: C.
32 À
1. Since
Z
Z
Z
Z
Z
2. 7 ´( r ) r *µ
~ 7 ´( µ b 7 ´) µ b 7 ´*µ c ²7 ´( q ) µ b 7 ´( q *µ b 7 ´) q *µ³ b 7 ´( q ) q *µ
~ À b À b À c ´²À³²À³ b ²À³²À³ b ²À³²À³µ b ²À³²À³²À³ ~ À .
If events ? and @ are independent, then so are ? and @ Á ? and @ , and ? and @ .
Answer: C
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
3. À ~ 7 ´* r * µ ~ 7 ´* µ b 7 ´* µ c 7 ´* q * µ ~ 7 ´* µ b 7 ´* µ c ²7 ´* µ³
Solving the quadratic equation results in 7 ´* µ ~ À (or À, but we disregard this
solution since 7 ´* µ must be  ). Alternatively, each of the five answers can be
substituted into the expression above for 7 ´* µ to see which one satisfies the equation.
Answer: B
4. 7 ´9µ ~ À Á 7 ´:µ ~ À Á 7 ´; µ ~ À .
7 ´9 q :µ ~ 7 ´
Á Á µ ~ À £ ²À³²À³ ~ 7 ´9µ h 7 ´:µ S 9Á : are not independent
7 ´9 q ; µ ~ 7 ´Á µ ~ À ~ ²À³²À³ ~ 7 ´9µ h 7 ´; µ S 9Á ; are independent
7 ´: q ; µ ~ 7 ´Jµ ~ £ ²À³²À³ ~ 7 ´:µ h 7 ´; µ S :Á ; are not independent. Answer: B
5. 7 ´, O, µ ~
7 ´, q, µ
7 ´, µ
~ S 7 ´, µ ~
, and similarly 7 ´, µ ~ 7 ´, µ ~
Then, 7 ´, r , r , µ
~ 7 ´, µ b 7 ´, µ b 7 ´, µ c ²7 ´, q , µ b 7 ´, q , µ b 7 ´, q , µ³
Answer: C
b 7 ´, q , q , µ ~ ² ³ c b .
6. 7 ´) received the letterO( did not receive an answer after writing to )µ
~
7 ´²)
received the letter)q(( did not receive an answer after writing to ) )µ
7 ´( did not receive an answer after writing to )µ
.
.
But, 7 ´( does not receive a reply after writing to ) ]
~ 7 ´²( does not receive a reply after writing to )³ q ²) received ('s letter³µ
b 7 ´²( does not receive a reply after writing to )³ q ²) did not receive ('s letter³µ
~ c
h b ~ c
.
7 ´() received the letter) q (( did not receive an answer after writing to ) )µ
~ c
h ~ c
.
7 ´²)
~
received the letter)q(( did not receive an answer after writing to ) )µ
7 ´( did not receive an answer after writing to )µ
²c³°
²c³°
~
c
c
.
Answer: C
7. Let * denote the event that a randomly chosen member of the population has a driver's
license and 5 is the event that the person reads a newspaper daily. Then
7 ´*µ ~ and 7 ´* r 5 µ ~ . Using probability rules we have
~ 7 ´* r 5 µ ~ 7 ´*µ b 7 ´5 µ c 7 ´* q 5 µ ~ b 7 ´5 µ c 7 ´*µ h 7 ´5 µ
(the last equality follows from the independence of * and 5 ), so that
~ b 7 ´5 µ c 7 ´5 µ S 7 ´5 µ ~ .
Answer: E
8. In the population of all accidents, + is the subset attributable to drunk drivers, and is the subset that are fatal. We are given 7 ´+O- µ ~ À Á 7 ´- µ ~ À Á 7 ´+µ ~ À .
c c
7 ´+q- µ
7 ´+q- µ
7 ´+q- µ
cc
c
We wish to find 7 ´+O- µ ~ 7 ´. However, À ~ 7 ´+O- µ ~ 7 ´- µ ~ À ,
µ
so that 7 ´+ q - µ ~ À . Then,
c c
7 ´+ q - µ ~ c 7 ´+ r - µ ~ c ²7 ´+µ b 7 ´- µ c 7 ´+ q - µ³ ~ À , so that
cc
7 ´+O- µ ~ À
~ À .
Answer: C
À