Download solutions

MATH 110 Sec 13.1 Intro to Probability Practice Exercises
We are flipping 3 coins and the outcomes are represented by
a string of H’s and T’s (HTH, etc.).
How many elements are there in the sample space?
Express the event “There are more heads than tails” as a set.
What is the probability that there are more heads than tails?
MATH 110 Sec 13.1 Intro to Probability Practice Exercises
We are flipping 3 coins and the outcomes are represented by
a string of H’s and T’s (HTH, etc.).
How many elements are there in the sample space?
By the Fundamental Counting Principle: 2 x 2 x 2 = 8
Express the event “There are more heads than tails” as a set.
{ HHH , HHT , HTH , THH }
What is the probability that there are more heads than tails?
By the Basic Probability Principle,
# 𝑤𝑎𝑦𝑠 𝑡𝑜 𝑔𝑒𝑡 𝑚𝑜𝑟𝑒 𝐻𝑒𝑎𝑑𝑠
4 1
𝑃 𝑚𝑜𝑟𝑒 𝐻 𝑡ℎ𝑎𝑛 𝑇 =
= =
# 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 8 2
MATH 110 Sec 13.1 Intro to Probability Practice Exercises
Four cards are drawn from a well-shuffled 52-card deck.
What is the probability of drawing a heart?
MATH 110 Sec 13.1 Intro to Probability Practice Exercises
Four cards are drawn from a well-shuffled 52-card deck.
What is the probability of drawing a heart?
13 DIAMONDS)
1
4 suits (CLUBS, SPADES, HEARTS,
𝑃 𝐻𝑒𝑎𝑟𝑡 =
52
13 CLUBS
13 SPADES
13 HEARTS
13 DIAMONDS
=
4
MATH 110 Sec 13.1 Intro to Probability Practice Exercises
Four cards are drawn from a well-shuffled 52-card deck.
What are the odds against drawing a heart?
Notice that there are 13 Hearts and 39 non-Hearts
Odds against drawing Heart are 39 : 13 which reduces to 3 : 1.
MATH 110 Sec 13.1 Intro to Probability Practice Exercises
Four cards are drawn from a well-shuffled 52-card deck.
What is the probability that all 4 are black?
MATH 110 Sec 13.1 Intro to Probability Practice Exercises
Four cards are drawn from a well-shuffled 52-card deck.
What is the probability that all 4 are black?
What is the probability that all 4 are black?
26 black cards
26 red cards
MATH 110 Sec 13.1 Intro to Probability Practice Exercises
Four cards are drawn from a well-shuffled 52-card deck.
What is the probability that all 4 are black?
From previous picture, there are
26 black cards and 52 cards in all.
Because the order the cards are drawn is not important, we
count the cards using combinations.
There are 𝐶 26,4 = 14950 ways to choose 4 black cards.
There are 𝐶 52,4 =270725 ways to choose any 4 cards.
By the Basic Probability Principle,
𝑃 𝑎𝑙𝑙 4 𝑏𝑙𝑎𝑐𝑘 =
# 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑡𝑜 𝑐ℎ𝑜𝑜𝑠𝑒 4 𝑏𝑙𝑎𝑐𝑘
# 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑡𝑜 𝑐ℎ𝑜𝑜𝑠𝑒 𝑎𝑛𝑦 4 𝑐𝑎𝑟𝑑𝑠
=
𝐶(26,4)
𝐶(52,4)
=
14950
270725
=
46
833
MATH 110 Sec 13.1 Intro to Probability Practice Exercises
Opinions of residents in a
town and surrounding area
about a proposed
racetrack is given here.
Support Oppose
Live in
town
Live in area
surrounding
3325
392
4747
617
A reporter randomly selects one of these 9081 people to interview.
What is the probability that the person is for the track?
What are the odds against the person supporting the track?
MATH 110 Sec 13.1 Intro to Probability Practice Exercises
Opinions of residents in a
town and surrounding area
about a proposed
racetrack is given here.
Support Oppose
Live in
town
Live in area
surrounding
3325
392
4747
617
A reporter randomly selects one of these 9081 people to interview.
What is the probability that the person is for the track?
3325 + 4747 8072 2 × 2 × 2 × 1009 8
𝑃 𝑆𝑢𝑝𝑝𝑜𝑟𝑡𝑠 𝑡𝑟𝑎𝑐𝑘 =
=
=
=
9081
9081
3 × 3 × 1009
9
What are the odds against the person supporting the track?
1:8
MATH 110 Sec 13.1 Intro to Probability Practice Exercises
If a dart is thrown and hits somewhere in the diagram
below, what is the probability that it hits the shaded area?
(Write final answer as a decimal rounded to 4 decimal places.)
8 in.
6 in.
4 in.
4 in.
3 in.
2 in.
2 in.
1 in.
MATH 110 Sec 13.1 Intro to Probability Practice Exercises
If a dart is thrown and hits somewhere in the diagram
below, what is the probability that it hits the shaded area?
(Write final answer as a decimal rounded to 4 decimal places.)
Note: The
probability of
hitting a region is
proportional to the
area of that region
and the whole
diagram.
8 in.
6 in.
4 in.
2 in.
2
4 in.
3 in.
1
2 in.
1 in.
2
𝑠ℎ𝑎𝑑𝑒𝑑 𝑎𝑟𝑒𝑎
2 × 1 + (2 × 2) 2 + 4
6
𝑃 ℎ𝑖𝑡 𝑠ℎ𝑎𝑑𝑒𝑑 𝑎𝑟𝑒𝑎 =
=
=
=
= 0.1875
𝑡𝑜𝑡𝑎𝑙 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑑𝑖𝑎𝑔𝑟𝑎𝑚
8×4
32
32
MATH 110 Sec 13.1 Intro to Probability Practice Exercises
If a dart is thrown and hits somewhere in the figure below
which is built from 4 different size squares (2 blue and 2
green), what is the probability that it hits the green area?
(Write final answer as an integer or simplified fraction.)
21 in.
9 in.
6 in.
15 in.
6 in.
15 in.
21 in.
9 in.
MATH 110 Sec 13.1 Intro to Probability Practice Exercises
If a dart is thrown and hits somewhere in the figure below
which is built from 4 different size squares (2 blue and 2
green), what is the probability that it hits the green area?
(Write final answer as an integer or simplified fraction.)
21 in.
9 in.
6 in.
6 in.
15 in.
21 in.
9 in.
15 in.
Note: The
probability of
hitting a
region is
proportional
to the area of
that region
and the
whole
diagram.
MATH 110 Sec 13.1 Intro to Probability Practice Exercises
If a dart is thrown and hits somewhere in the figure below
which is built from 4 different size squares (2 blue and 2
green), what is the probability that it hits the green area?
(Write final answer as an integer or simplified fraction.)
9 in.
9 in.
6 in.
15 in.
21 in.
225
6 in.
15 in.
21 in.
15 in.
Area of
the biggest
green
square is
15 x 15 = 225
15 in.
Note: The
probability of
hitting a
region is
proportional
to the area of
that region
and the
whole
diagram.
MATH 110 Sec 13.1 Intro to Probability Practice Exercises
If a dart is thrown and hits somewhere in the figure below
which is built from 4 different size squares (2 blue and 2
green), what is the probability that it hits the green area?
(Write final answer as an integer or simplified fraction.)
- 81
144
9 in.
Subtract
blue area
b/c it
covers up
part of the
green.
9 in.
6 in.
15 in.
21 in.
225
6 in.
9 in.
15 in.
9 in.
Area of
Small blue
square is
9 x 9 =81
21 in.
15 in.
Area of
the biggest
green
square is
15 x 15 = 225
15 in.
Note: The
probability of
hitting a
region is
proportional
to the area of
that region
and the
whole
diagram.
MATH 110 Sec 13.1 Intro to Probability Practice Exercises
If a dart is thrown and hits somewhere in the figure below
which is built from 4 different size squares (2 blue and 2
green), what is the probability that it hits the green area?
(Write final answer as an integer or simplified fraction.)
144 + 36 = 180
9 in.
9 in.
- 81
144
6 in.
The small
green square
sits on the
small blue
square and
adds back
more green.
6 in.
Subtract
blue area
b/c it
covers up
part of the
green.
+ 36
6 in.
15 in.
21 in.
225
36
15 in.
9 in.
15 in.
9 in.
Area of
Small blue
square is
9 x 9 =81
21 in.
6 in.
Area of
the biggest
green
square is
15 x 15 = 225
15 in.
Note: The
probability of
hitting a
region is
proportional
to the area of
that region
and the
whole
diagram.
MATH 110 Sec 13.1 Intro to Probability Practice Exercises
the total
is 180 square
inches.
If a dart is So,
thrown
andgreen
hits area
somewhere
in the
figure below
The total
area of
the whole
figure is
21 xsquares
21 = 441(2
square
which
is built
from
4 different
size
blue inches.
and 2
green), what is the probability that it hits the green area?
(Write final answer as an integer or simplified fraction.)
144 + 36 = 180
9 in.
9 in.
- 81
144
6 in.
The small
green square
sits on the
small blue
square and
adds back
more green.
6 in.
Subtract
blue area
b/c it
covers up
part of the
green.
+ 36
6 in.
15 in.
21 in.
225
36
15 in.
9 in.
15 in.
9 in.
Area of
Small blue
square is
9 x 9 =81
21 in.
6 in.
Area of
the biggest
green
square is
15 x 15 = 225
15 in.
Note: The
probability of
hitting a
region is
proportional
to the area of
that region
and the
whole
diagram.
MATH 110 Sec 13.1 Intro to Probability Practice Exercises
the total
is 180 square
inches.
If a dart is So,
thrown
andgreen
hits area
somewhere
in the
figure below
The total
area of
the whole
figure is
21 xsquares
21 = 441(2
square
which
is built
from
4 different
size
blue inches.
and 2
green), what is the probability that it hits the green area?
(Write final answer as an integer or simplified fraction.)
144 + 36 = 180
9 in.
9 in.
- 81
144
6 in.
The small
green square
sits on the
small blue
square and
adds back
more green.
6 in.
Subtract
blue area
b/c it
covers up
part of the
green.
+ 36
21 in.
225
36
15 in.
9 in.
15 in.
9 in.
Area of
Small blue
square is
9 x 9 =81
21 in.
6 in.
Area of
the biggest
green
square is
15 x 15 = 225
15 in.
Note: The
probability of
hitting a
region is
proportional
to the area of
that region
and the
whole
diagram.
6 in.
15 in.
180 20
𝑃 ℎ𝑖𝑡𝑠 𝑔𝑟𝑒𝑒𝑛 =
=
441 49
MATH 110 Sec 13.1 Intro to Probability Practice Exercises
Player 1 & Player 2 play a ame using
Spinner A and Spinner B as shown.
Player 1 gets to choose one of the
spinners, both players spin, and the
one getting the larger number wins.
Which spinner should Player 1 choose?
Assuming that choice of spinner what is the probability
that Player 1 wins?
MATH 110 Sec 13.1 Intro to Probability Practice Exercises
Player 1 & Player 2 play a ame using
Spinner A and Spinner B as shown.
Player 1 gets to choose one of the
spinners, both players spin, and the
one getting the larger number wins.
List every possible way the 2 spinners could land, then count the # of times each wins.
Which spinner should Player 1 choose?
Assuming that choice of spinner what is the probability
that Player 1 wins?
MATH 110 Sec 13.1 Intro to Probability Practice Exercises
Player 1 & Player 2 play a ame using
Spinner A and Spinner B as shown.
Player 1 gets to choose one of the
spinners, both players spin, and the
one getting the larger number wins.
List every possible way the 2 spinners could land, then count the # of times each wins.
Which spinner should Player 1 choose?
A spin
1
1
1
4
4
4
9
9
9
B spin
2
7
8
2
7
8
2
7
8
Who wins
B
B
B
A
B
B
A
A
A
Assuming that choice of spinner what is the probability
that Player 1 wins?
MATH 110 Sec 13.1 Intro to Probability Practice Exercises
Player 1 & Player 2 play a ame using
Spinner A and Spinner B as shown.
Player 1 gets to choose one of the
spinners, both players spin, and the
one getting the larger number wins.
List every possible way the 2 spinners could land, then count the # of times each wins.
Which spinner should Player 1 choose? B (wins 5 out of 9 times)
A spin
1
1
1
4
4
4
9
9
9
B spin
2
7
8
2
7
8
2
7
8
Who wins
B
B
B
A
B
B
A
A
A
Assuming that choice of spinner what is the probability
that Player 1 wins?
MATH 110 Sec 13.1 Intro to Probability Practice Exercises
Player 1 & Player 2 play a ame using
Spinner A and Spinner B as shown.
Player 1 gets to choose one of the
spinners, both players spin, and the
one getting the larger number wins.
List every possible way the 2 spinners could land, then count the # of times each wins.
Which spinner should Player 1 choose? B (wins 5 out of 9 times)
A spin
1
1
1
4
4
4
9
9
9
B spin
2
7
8
2
7
8
2
7
8
Who wins
B
B
B
A
B
B
A
A
A
Assuming that choice of spinner what is the probability
5
that Player 1 wins?
𝑃 𝐵 𝑤𝑖𝑛𝑠 =
9
MATH 110 Sec 13.1 Intro to Probability Practice Exercises
Some more detailed solutions and some more problems
and solutions can be found here:
http://cas.ua.edu/mtlc/UAMath110/Exercises/Sec13-1ExercisesSOL.pdf