ACT245H1 S - PROBLEM SET 5
1. A class contains 8 boys and 7 girls. The teacher selects 3 of the children at random and
without replacement. Calculate the probability that number of boys selected exceeds the
number of girls selected.
A) B) B) B) B) 2. There are 97 men and 3 women in an organization. A committee of 5 people is chosen
at random, and one these 5 is randomly designated as chairperson. What is the
probability that the committee includes all 3 women and has one of the women as
chairperson?
²[ [³
A) ²[³
[ [
B) ²[³
²[ [³
C) ²[³
[ [
D) [ [
E) 3. A box contains 4 red balls and 6 white balls. A sample of size 3 is drawn without
replacement from the box. What is the probability of obtaining 1 red ball and 2 white
balls, given that at least 2 of the balls in the sample are white?
A) B) C) D) E) 4. When sent a questionnaire, 50% of the recipients respond immediately. Of those who
do not respond immediately, 40% respond when sent a follow-up letter. If the
questionnaire is sent to 4 persons and a follow-up letter is sent to any of the 4 who do not
respond immediately, what is the probability that at least 3 never respond?
A) ²À³ b ²À³ ²À³
B) ²À³ ²À³
C) ²À³ b ²À³ ²À³
D) À²À³²À³ b ²À³
E) ²À³ b ²À³ ²À³
5. A box contains 35 gems, of which 10 are real diamonds and 25 are fake diamonds.
Gems are randomly taken out of the box, one at a time without replacement. What is the
probability that exactly 2 fakes are selected before the second real diamond is selected?
$ %$ %
4
A) B) C)
D) 6 7² E) 6 7² ³ ² ³
³ ² ³
$ %
6. A fair coin is tossed. If a head occurs, 1 fair die is rolled; if a tail occurs, 2 fair dice
are rolled. If @ is the total on the die or dice, then 7 ´@ ~ µ ~
A) B) C) D) E) 7. Three boxes are numbered 1, 2 and 3. For ~ Á Á , box contains blue marbles
and c red marbles. In a two-step experiment, a box is selected and 2 marbles are
drawn from it without replacement. If the probability of selecting box is proportional to
, what is the probability that the two marbles drawn have different colors?
A) B) C) D) E) 8. A number is ? chosen at random from the series 2,5,8, ... and another number @ is
chosen at random from the series 3,7,11, ... Each series has 100 terms. Find 7 ´? ~ @ µ.
A) À
B) À
C) À
D) À
E) À
ACT245H1 S - PROBLEM SET 5 SOLUTIONS
[
1. There are 8 9 ~ [h[
~ hh
hh ~ ways of selecting children from a
group of without replacement. The number of boys selected exceeds the number of
girls selected if either
(i) boys and girls are selected, or (ii) boys and girl are selected .
[
[
There are 8 9 h 8 9 ~ [h[
h [h[
~ ways in which selection (i) can occur , and
[
[
there are 8 9 h 8 9 ~ [h
[
h [h
[
~ ways in which selection (ii) can occur.
The probability of either (i) or (ii) occurring is b
Answer: E
~ À
2. Let ( be the event that the committee has a woman as chairperson, and let ) be the
event that the committee includes all 3 women. Then, 7 ´( q )µ ~ 7 ´(O)µ h 7 ´)µ .
The conditional probability 7 ´(O)µ is equal to since the chairperson is chosen at
random from the 5 committee members, and, given ) , 3 of the committee members are
women. There are 6 7 ways of choosing a 5-member committee from the group of
100. Out of all 5-member committees, there are 6 7 committees that include all 3
women (i.e., 2 men are chosen from the 97 men). Thus,
$ %
[ [
, [
7 ´)µ ~ ~ ² [[
[ ³ ² [ [ ³ ~ [ [ ,
$ %
[ [
and 7 ´( q )µ ~ [[[
h ~ h[
[ [ .
3. 7 ´9Á > Oat least > µ ~
7 ´9Á> µ
7 ´at least > µ
Answer: A
~
$ 41 %$ 62 %
$ 41 %$ 62 %b$ 40 %" 63 #
~ . Answer: C
4. The probability that an individual will not respond to either the questionnaire or the
follow-up letter is ²À³²À
³ ~ À . The probability that all 4 will not respond to either the
questionnaire or the follow-up letter is ²À³ .
7 ´ don't respondµ ~ 7 ´ response on 1st round, no additional responses on 2nd roundµ
b 7 ´no responses on 1st round, 1 response on 2nd roundµ
~ ´²À³ ²À
³ µ b ´²À³ ²À
³ ²À³µ ~ ²À³ ²À³ . Then,
Answer: A
7 ´at least 3 don't respondµ ~ ²À³ b ²À³ ²À³ .
5. Exactly 2 fakes must be picked in the first 3 picks and the second real diamond must
occur on the 4th pick. The possible ways in which this may occur are (- -fake, 9 -real)
hhh
hhh
hhh
- - 99 (prob. hhh
) , - 9- 9 (prob. hhh
) , 9- - 9 (prob. hhh
).
hhh
The overall probability is h hhh
~ .
Answer: B
6. If 1 fair die is rolled, the probability of rolling a 6 is , and if 2 fair dice are rolled, the
probability of rolling a 6 is (of the 36 possible rolls from a pair of dice, the rolls 1-5,
2-4, 3-3, 4-2 and 5-1 result in a total of 6), Since the coin is fair, the probability of rolling
a head or tail is .5. Thus, the probability that @ ~ is ²À³² ³ b ²À³² ³ ~ À
Answer: C
7. If the probability of selecting box 1 is , then b b ~ S ~ .
Then the probability in question is
7 ´2 different colorsObox 1 selectedµ h 7 ´box 1 selectedµ
b 7 ´2 different colorsObox 2 selectedµ h 7 ´box 2 selectedµ
b 7 ´2 different colorsObox 3 selectedµ h 7 ´box 3 selectedµ
~ h
h b h
h b h
h ~ h
~ . Answer: E
$ % $ % $ % 8. There are ~ Á equally likely possible choices of ²?Á @ ³. Of these choices,
the pairs that equal ? and @ are ²Á ³-1 Á ²Á ³-2 Á ²Á ³-3 Á À À À Á ²Á ³-25
(they are of the form ² c Á c ³. The probability is Á
. Answer: A