Section 4.3 Solutions.
1) If two dates are selected at random from the 365 days of the year and are listed in the order
they were selected.
a) Use the counting principle to determine the number of possible outcomes if the dates are
selected with replacement.
_____365____*_____365_____= 133225
First day
second day
Answer: 133,225 possible outcomes
b) Use the counting principle to determine the number of possible outcomes if the dates are
selected without replacement.
_____365____*_____364_____= 132860
First day
second day
Answer: 132,860 possible outcomes
3) Joe has 5 shirts, 3 pairs of pants and 4 pairs of shoes. He needs to make an outfit containing
one of each item. How many different outfits are possible? (We assume that one different
item makes a different outfit)
__5_____*____3____*___4_____ = 60
shirts
pants
shoes
Answer: 60 outfits
5) A television remote has buttons for the digits 0 – 9. If you press two buttons, how many
numbers are possible if
a) The same button may be pressed twice?
This is like a with replacement problem.
_____10____*_____10_____= 100
First number
second number
Answer: 100 possible outcomes
b) The same button may not be pressed twice?
This is like a without replacement problem.
_____10____*_____9_____= 90
First number
second number
Answer: 90 possible outcomes
7) A movie theater sells 3 sizes of popcorn (small, medium, and large) with 3 choices of
toppings (no butter, butter, extra butter). How many possible ways can a bag of popcorn be
purchased?
______3______*_____3______ = 9
Size
topping
Answer: 9
9) A state issues license plates consisting of letters and numbers. There are 26 letters and the
letters may be repeated. There are 10 digits and the digits may be repeated. How many
possible license plates can be issued with two letters followed by three numbers?
____26____*______26______*____10______*____10_______*____10______= 676,000
First letter second letter
1st number
2nd number
3rd number
Answer: 676,000
11) Burger Queen offers 4 types of burgers, 5 types of beverages, and 3 types of desserts. If a
meal consists of 1 burger, one beverage and one dessert, how many possible meals can be
chosen?
____4________*_____5_______*_____3_____= 60
Burger
beverage
dessert
Answer: 60
13) Reid is ordering a birthday cake for a friend. There are 5 cake flavors and 4 frosting flavors
to choose from. How many different cakes can Reid order?
_____5_____*____4_____ = 20
Cake
frosting
Answer: 20
15) Answer question 14 but assume the license plates must be of the form 2 letters followed
by 4 numbers and there are no other restrictions.
____26___*______26___*____10______*____10_______*____10___*__10_____= 6,760,000
First letter
second letter
1st number
2nd number 3rd number 4th number
Answer: 6,760,000
17) The standard New York state license plate has three letters followed by four digits. How
many different license plates are possible if neither the digits nor the letters can be repeated?
____26___*______25___*____24_____*___10____*____9___*____8___*__7_____
1st letter
2nd letter
3rd letter
1st #
2nd #
3rd #
4th #
Answer: 78,624,000
19) You are assigned a computer generated 4-digit password to access your new voice mail
account. If the digits can be repeated, how many passwords are possible?
___10____*___10__*____10___*___10____
1st #
2nd #
3rd #
4th #
Answer: 10,000
#20 – 30: Counting principle problems with sample space construction and probability
questions.
21) A couple plans to have two children.
a) Determine the number of points in the sample space of the possible arrangements of
boys and girls.
_______2_____*______2________ = 4
First child
second child
Answer: 4
b) Construct a tree diagram and list the sample space.
Answer: Sample Space: {BB BG GB GG}
Find the probability the couple has:
c)
Two girls
The numerator will be the one element in the sample space that has two girls.
The denominator will be the 4 elements in the sample space.
Answer: 1/4
d) At least one girl
The numerator counts BG GB and GG, it should be 3.
The denominator consists of all 4 elements in the sample space.
Answer: 3/4
e) A girl then a boy
The numerator is 1, as GB is the only element that I count here.
The denominator is still the 4 elements in the sample space.
Answer: ¼
23) A coin is tossed three times and a sequence of heads and tails is recorded.
a) Determine the number of points in the sample space
____2_____*_____2_____*_____2______= 8
1st flip
2nd flip
3rd flip
Answer: 8
b) Construct a tree diagram and list the sample space
Answer: Sample Space {HHH HHT HTH HTT THH THT TTH TTT}
Find the probability that:
c)
No heads are tossed
The numerator is 1 as the TTT is the only event with no heads.
The denominator is the 8 items in the sample space.
Answer: 1/8
d) Exactly one head is tossed
The numerator consists of these 3 elements: TTH THT HTT
The denominator is the 8 items in the sample space.
Answer: 3/8
e) Three heads are tossed
The numerator is 1 as the HH is the only event with three heads.
The denominator is the 8 items in the sample space.
Answer: 1/8
25) Two six sided dice are rolled and the number on each face is recorded.
a) Determine the number of points in the sample space
_______6_______*_______6______= 36
first dice
second dice
Answer: 36
b) Construct a tree diagram and list the sample space
25b) Sample Space:
(1,1) sum 2
(2,1) sum 3
(3,1) sum 4
(4,1) sum 5
(5,1) sum 6
(6,1) sum 7
(1,2) sum 3 (1,3) sum 4 (1,4) sum 5 (1,5) sum 6 (1,6) sum 7
(2,2) sum 4 (2,3) sum 5 (2,4) sum 6 (2,5) sum 7 (2,6) sum 8
(3,2) sum 5 (3,3) sum 6 (3,4) sum 7 (3,5) sum 8 (3,6) sum 9
(4,2) sum 6 (4,3) sum 7 (4,4) sum 8 (4,5) sum 9 (4,6) sum 10
(5,2) sum 7 (5,3) sum 8 (5,4) sum 9 (5,5) sum 10 (5,6) sum 11
(6,2) sum 8 (6,3) sum 9 (6,4) sum 10 (6,5) sum 11 (6,6) sum 12
c) Find the probability that a double is rolled (both dice have the same number)
The numerator is 6. I need to count these: (1,1) (2,2) (3,3) (4,4) (5,5) (6,6)
The denominator is the 36 items in the sample space.
Answer: 6/36 = 1/6
d) Find the probability that a sum of 7 is rolled
The numerator is 6. I need to count: (1,6) (2,5) (3,4) (4,3) (5,2) (6,1)
The denominator is the 36 items in the sample space.
Answer: 6/36 = 1/6
e) Find the probability a sum of 2 is rolled
The numerator is a 1 as the only item I can count is (1,1)
The denominator is the 36 items in the sample space.
Answer: 1/36
f)
Are you as likely to roll a sum of 2 as you are of rolling a sum or 7
Answer: It is 6 times more likely to roll the sum of a 7.
27) A coin is flipped then a 6 sided dice is rolled.
a) Determine the number of points in the sample space
_____2______*______6______= 12
Coin
dice
b) Construct a tree diagram and list the sample space
Sample Space: { H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6}
c) Find the probability that a head is flipped and an even number is rolled
The numerator counts: H2 H4 H6, The numerator will be 3.
The denominator counts the 12 items in the sample space.
Answer: 3/12 = ¼
d) Find the probability that a tail is flipped and a number less than 3 is rolled
The numerator counts T1 T2, the numerator is 2.
The denominator counts the 12 items in the sample space.
Answer: 2/12 = 1/6
29) A coin is flipped then a number is picked out of a hat containing the numbers 1,2 and 3.
a) Determine the number of points in the sample space
____2_______*_____3______ = 6
Coin
number
Answer: 6
b) Construct a tree diagram and list the sample space
Sample Space: {H1 H2 H3 T1 T2 T3}
c)
Find the probability that a head is flipped and an even number is selected
The numerator is 1, as I can only count H1.
The denominator is the 6 elements in the sample space.
Answer: 1/6
d) Find the probability that a tail is flipped and a number less than 3 is selected
The numerator is 2. I can count T1 and T2
The denominator is the 6 elements in the sample space.
Answer: 2/6 = 1/3