Probability Practice Quest 2
Solution Guide
1. Two cards are chosen at random from a deck of 52 cards without replacement. What is the
probability of getting two kings?
4/52 · 3/51 = 1/221
2. In a shipment of 100 televisions, 6 are defective. If a person buys two televisions, what is
the probability that both are defective if the first television is not replaced after it is
purchased?
6/100 · 5/99 =1/330
On a math test, 5 out of 20 students got an A. If three students are chosen at random without
replacement, what is the probability that all three got an A on the test?
5/20 · 4/19 · 3/18 =1/114
Three cards are chosen at random from a deck of 52 cards without replacement. What is the
probability of getting an ace, a king and a queen in order?
4/52 · 4/51 · 4/50 = 8/16575
A school survey found that 7 out of 30 students walk to school. If four students are selected at
random without replacement, what is the probability that all four walk to school?
7/30 · 6/29 · 5/28 · 4/27 = 1/783
At a middle school, 18% of all students play football and basketball, and 32% of all students
play football. What is the probability that a student who plays football also plays basketball?
P( BB/ FB) =
P(BB F B)/P(FB) = .
18/32 = 9/16
In Europe, 88% of all households have a television. 51% of all households have a television and
a VCR. What is the probability that a household with a television also has a VCR?
P(VCR/ TV) =
P(VCR  TV/P(TV) =
51/88
A research organization mailed out questionnaires to 10
random people. If the probability that any one person will
answer the questionnaire is 10%, find the probability that at
least 8 will answer
n = 10, k = 8, 9, 10, p = 10%, 1 - p = 90%
k(1-p)n-k
C
p
n k
8(.9)2 +
C
(.1)
10 8
9(.9)1 +
C
(.1)
10 9
10
C
(.1)
10 10
11)Of 200 students surveyed, 100 said
they studied for their math test. 105 said
they passed their test, and 80 said they
studied and passed their test.
a.
Incorporate the facts
given above into a
conditional chart.
P P
S
S
11)Of 200 students surveyed, 100 said
they studied for their math test. 105 said
they passed their test, and 80 said they
studied and passed their test.
a.
Incorporate the facts
given above into a
conditional chart.
P P
S
S
80
100
105
200
11)Of 200 students surveyed, 100 said
they studied for their math test. 105 said
they passed their test, and 80 said they
studied and passed their test.
P P
S
S
80
105
P P
100
200
S
S
80 20 100
25 75 100
105 95 200
11)Of 200 students surveyed, 100 said
they studied for their math test. 105 said
they passed their test, and 80 said they
studied and passed their test.
a) P(pass)
105
200
21
40
P P
S
S
80 20 100
25 75 100
105 95 200
11)Of 200 students surveyed, 100 said
they studied for their math test. 105 said
they passed their test, and 80 said they
studied and passed their test.
b) P(pass/studied) P(P/S)
80
100
4
5
P P
S
S
80 20 100
25 75 100
105 95 200
11)Of 200 students surveyed, 100 said
they studied for their math test. 105 said
they passed their test, and 80 said they
studied and passed their test.
_ _
c) P(P/S)
75
100
3
4
P P
S
S
80 20 100
25 75 100
105 95 200
In a throw of a red die, r, and a white
die, w, find:
 P (r = 5, given that
r + w > 9)
(1, 1)
(1,2)
(1, 3)
(1, 4)
(1, 5)
(1, 6)
(2, 1)
(2, 2)
(2, 3)
(2, 4)
(2, 5)
(2, 6)
(3, 1)
(3, 2)
(3, 3)
(3, 4)
(3, 5)
(3, 6)
(4, 1)
(4, 2)
(4, 3)
(4, 4)
(4, 5)
(4, 6)
(5, 1)
(5, 2)
(5, 3)
(5, 4)
(5, 5)
(5, 6)
(6, 1)
(6, 2)
(6, 3)
(6, 4)
(6, 5)
(6, 6)
P (r = 5 | r + w > 9)
In a throw of a red die, r, and a white
die, w, find:
 P (r = 5, given that
r + w > 9)
(3, 6)
(4, 5)
(4,6)
(5, 4)
(5,5)
(5,6)
(6,4)
(6,5)
(6,6)
P (r = 5 | r + w > 9)
3
10
(6, 3)
In a throw of a red die, r, and a white
die, w, find:
P (r + w = 9, given
that
w = 6)
P (r + w = 9 |w = 6)
(1, 1)
(1,2)
(1, 3)
(1, 4)
(1, 5)
(1, 6)
(2, 1)
(2, 2)
(2, 3)
(2, 4)
(2, 5)
(2, 6)
(3, 1)
(3, 2)
(3, 3)
(3, 4)
(3, 5)
(3, 6)
(4, 1)
(4, 2)
(4, 3)
(4, 4)
(4, 5)
(4, 6)
(5, 1)
(5, 2)
(5, 3)
(5, 4)
(5, 5)
(5, 6)
(6, 1)
(6, 2)
(6, 3)
(6, 4)
(6, 5)
(6, 6)
In a throw of a red die, r, and a white
die, w, find:
P (r + w = 9, given that
w = 6)
(1, 6)
P (r + w = 9 |w = 6)
(3, 6)
(2, 6)
(4, 6)
1
6
(5, 6)
(6, 6)
P(J  K)
With Replacement
4 4
52 52
1  1
13 13
1_
169
Without Replacement
4 4
52 51
1  4
13 51
4_
663
There are 8 females and 5 males in my class. Half of the females are
adults and 2 of the males are adults. If I randomly select 4 people (2
adults and 2 students) to go to the board, find P(all females). Leave in
combination notation.
C
•
C
•
C
•
C
4 2 2 0 4 2 3 0
C
13 4
(3x + 2y)4
1( )4 + 4( )3( )1+ 6( )2( )2+ 4( )1( )3+ 1( )4
1(3x)4 + 4(3x)3(2y)1+ 6(3x)2(2y)2+ 4(3x)1(2y)3+ 1(2y)4
1(81x4) + 4(27x3)(2y) + 6(9x2)(4y2)+ 4(3x)(8y3)+1(16y4)
81x4 + 216x3y) + 216x2y2 + 96xy3 + 16y4
fourth term of (2x –
1( )5 + 5( )4( )1 + 10( )3( )2 + 10( )2( )3
10(2x)2(-3y)3
10(4x2)(-27y3)
-1080x2y3
5
3y)
What is the probability of getting exactly 2 fives in 4 rolls of a die?
n = 4, k =2, P = 1/6, 1-P = 5/6
k(1-p)n-k
C
p
n k
2
2
4C2(1/6) (5/6) =
6 (1/36)(25/36) =
25/216
15)A dollar bill changer was tested with 100
one-dollar bills, of which 25 were counterfeit
and the rest were legal. The changer rejected
30 bills, and 6 of the rejected bills were
counterfeit. Construct a table, then find
a.
Incorporate the facts
given above into a
conditional chart.
R R
C
C
15)A dollar bill changer was tested with 100
one-dollar bills, of which 25 were counterfeit
and the rest were legal. The changer rejected
30 bills, and 6 of the rejected bills were
counterfeit. Construct a table, then find
a.
Incorporate the facts
given above into a
conditional chart.
R R
C
C
6
25
30
100
15)A dollar bill changer was tested with 100
one-dollar bills, of which 25 were counterfeit
and the rest were legal. The changer rejected
30 bills, and 6 of the rejected bills were
counterfeit. Construct a table, then find
R R
C
C
6
30
R R
25
100
C
C
6
19 25
24 51 75
30 70 100
15)A dollar bill changer was tested with 100
one-dollar bills, of which 25 were counterfeit
and the rest were legal. The changer rejected
30 bills, and 6 of the rejected bills were
counterfeit. Construct a table, then find
a.
b.
c.
The probability that a bill is legal
and it is accepted by the
machine.
P(CR)= 51
100
The probability that a bill is
rejected, given it is legal.
P(R/C)= 24 = 8
75 25
The probability that a
counterfeit bill is not rejected
P(R/C) = 19
25
R R
C
C
6
19 25
24 51 75
30 70 100
A quiz has 6 multiple-choice questions, each with 4
choices. What is the probability of getting at least 5
of 6 questions correct?
n = 6, k >5, P = .25, 1- P = .75
k(1-p)n-k
C
p
n k
5
6
6C5(.25) (.75) + 6C6(.25)
According to the National Institute of Health, 32% of all
women will suffer a hip fracture due to osteoporosis by the
age of 90. Six women aged 90 are randomly selected, find
the probability that exactly 3 will have suffered a hip fracture
at most 3 of them will have
at least 1 of them will have
suffered a hip fracture
suffered a hip fracture.
n = 6, k = < 3, p = 32%,
1 – 6C0 (.68)6
1 – p = 68%
k
n-k
nCk p (1-p)
3 (.68)3 +
C
(.32)
6 3
2
4
6C2 (.32) (.68) +
1
5
6C1 (.32) (.68) +
6
C
(.68)
6 0