Casino 2
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Casino Games Activity 2
8
Remember: • SHOW ALL WORK. You must communicate to me how your got your solution. This is as important as your solution. • Show your work and your answers on a separate sheet at the end, then you only need to submit that sheet (not these sheets). In the last activity, you found simple probabilities. In this activity, we’ll look at some probabilities that are more complex. We’ll also see how understanding the mathematical concept of sets can help you.
Sets What is a set? In probability, a set is a group of objects with a set of rules to decide if an object belongs with the group or not. “Everyone enrolled in this course” is a set. It’s a group of objects (people), and I can tell who’s in the set (you) and who’s not (President Obama). “Young adults” is not a set unless it’s better defined who is a young adult and who is not. Example 1: Write the elements belonging to each of the following sets. set: natural numbers less than 5 Answer: {1,2,3,4} set: states that border Minnesota Answer: {Iowa, Wisconsin, North Dakota, South Dakota}
What is a subset? A subset is a set completely contained within another set. Example 2: Tell if B is a subset of A. A: natural numbers less than 5 B: {4,5} Answer: no, because 5 is not part of A A: the 50 United States of America B: states that border Minnesota Answer: yes, each of the states that borders Minnesota is a member of the USA
Problem Set 3 1. Answer yes or no if a-f are elements of the set. set: {1,2,3,4,5,6,7,8,9} for example: i. 7 yes
ii. 99 no
a. 2
e. 12
b. 3
c. 10
d. 0
f. 4
2. Answer yes or no if a-e are subsets of the set set: Big Ten schools {Illinois, Indiana, Iowa, Michigan, Michigan State, Minnesota, Northwestern, Ohio State, Penn State, Purdue, Wisconsin} a. {Michigan, Michigan State}
b. {Minnesota}
c. {Iowa, Iowa State}
d. {Indiana State}
e. {Ohio State, Michigan State, Penn State}
Probability What are mutually exclusive events? In probability, mutually exclusive events are two events that can not occur at the same time, such as getting both heads and tails on the same coin flip, or having a student tell you she is in 9th grade and 12th grade at the same time. Example 1: Are the events mutually exclusive? event 1: rolling a single die and getting a 4,5, or 6
event 2: rolling a single die and getting a 1 or 2
yes, these events are mutually exclusive because if you get a 4,5, or 6, that means you didn’t get a 1 or 2 event 1: choosing a random student who is female
event 2: choosing a random student who is in 9th grade
no, these events are not mutually exclusive because you could be a girl in 9th grade
Some probability question ask the probability of event A “OR” event B. To find these probabilities, simply add the probability of A plus the probability of B if A and B are mutually exclusive events. Example 2: What is the probability of drawing from a deck of cards a 2 or a queen. These are mutually exclusive events—a card is never both a 2 and a queen. The probability of drawing a 2 is 4/52. The probability of drawing a queen is also 4/52. 4/52 + 4/52 = 8/52 = 2/13. So the probability is 2/13.
What if the events are not mutually exclusive? You still add the two probabilities of the two separate events, but there’s one more step—subtracting the probability of both. Example 3: If a single card is drawn from a deck of 52, find the probability that it will be a red or a face card. P(red card) = 26 red cards/52 total cards P(face card) = 12 face card/52 total cards P(red) + P(face) = 38/52 P(red and face) = 6 red face cards/52 total cards P(red or face) = 38/52 - 6/52 = 32/52 = 8/13
Problem Set 4 Answer if the events in 1-6 are mutually exclusive (m.e.) or not mutually exclusive 1. owning a car and owning a truck 2. wearing glasses and wearing sandals 3. being married and being over 30 years old 4. being a teenager and being over 30 years old 5. getting a 4 and an odd number on a die you rolled one time 6. being in 12th grade and not being in 12th grade Find the probabilities of drawing the following cards from a standard deck 7. less than a 4 (count aces as ones) 8. a black card or an ace 9. a heart or a jack 10. a red card or a seven
Graphing and Data Analysis What is a bar graph? When we’re keeping track of different categories of data, it can be useful to show how much of the data is in what categories with a bar graph.
8
8
8
Example 1: Each bar represents a number of students who received a bachelor’s degree in different major areas. If twice as many people receive a degree is business as in the social sciences, the business bar will be twice as long as the social sciences bar. One way this could be different from a circle graph is that there might be other disciplines (foreign language, for example) that aren’t shown on this graph. (If you printed this out in black and white, you may want to check out the color coding in the original pdf.)
What is a circle graph? When we’re keeping track of different categories of data, it can be useful to show how much of the data is in what categories with a circle graph, too. It shows proportions, like a bar graph, but in a slightly different way. Example 2: The entire circle represents ALL U.S. public school students, so every student needs to belong in one slice of the pie.
What is a dot plot? A dot plot keeps track of frequency, or how often, different results show up. This is useful when dealing with small amounts of quantitative (numerical) data. Example 3: Each dot represents the population for one of the 50 U.S. states. Minnesota is in red. When different states have the same population, the dot stack up on top of each other.
What is a stemplot? A stemplot is a simple way to visually represent small amounts of quantitative (number) data. This is sometimes called a stem and leaf plot because each piece of data is separated into stem and leaf pieces. Example 4: This table shows student scores on a final exam. Take the first digit in each two digit score, and have it represent the stem. The second digit will be the leaf. In this way, the graph will turn out a little like a dotplot.
student score
A
B
C
D
E
F
G
H
I
J
64 87 92 81 88 73 77 75 80 70
6 4 7 0357 8 0178 9 2
What is a histogram? A histogram is a visual display similar to a stemplot, but it can be used with large data sets. A histogram has two dimension. One is the range of values, divided into equal subunits, and the other is the frequency of data values in those ranges. Example 5: Use the student exam score data above to create a histogram.
What is a scatterplot? A scatterplot compares two quantitative values simultaneously. On a two-dimensional graph, the x-axis axis is labeled with one variable, and the y-axis is labeled with the other. Example 6: Create a scatterplot comparing the variables “mpg” and “weight.” Notice that each dot represents one car.
Problem Set 5 Look at the bar graph to the right. 1. What do the light and dark bars represent? 2. What are the numbers on top of the columns? How do you know? 3. What is the difference between the total math score in 1967 and 1999?
The graph to the right is the same thing as a circle graph, even though it’s not in the shape of a circle. We might call it an area graph. 4. What was the most sold book type in the U.S. according to the graph? 5. Which is more popular, book club books or college books?
Make your own dot plot. In the table, some data is compiled regarding the number of states visited by Sobriety High School students. Add the number of states you’ve visited to the table. student
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
p
number of states visited
5
11
11
14
28
6
6
9
12
18
5
10
4
14
5
8
you
states visited 6. Create a dot plot with the data in the table. You can use this graph here, or re-create it on your own answer sheet. 7. Circle the dot on the dotplot that represents you. 8. What looks to be about the average number of states visited? (you don’t have to be exact, just give a guess that’s close by looking at the dotplot.)
Use the data in the table describing the lengths of reigns of British rulers beginning with William the Conqueror in 1066 to create a stem plot. 9. How long was the longest reign? Who ruled the longest? 10. Create a stemplot from the lengths of reigns. Use the numerals 0-6 on the left of the line. 0 = 00.
ruler
reign
ruler
reign
ruler
reign
ruler
reign
William I
21
Edward III
50
Edward VI
6
George I
13
William II
13
Richard II
22
Mary I
5
George II
33
Henry I
35
Henry IV
13
Elizabeth I
44
George III
59
Stephen
19
Henry V
9
James I
22
George IV
10
Henry II
35
Henry VI
39
Charles I
24
William IV
7
Richard I
10
Edward IV
22
Charles II
25
Victoria
63
John
17
Edward V
0
James II
3
Edward VII
9
Henry III
56
Richard III
2
William III
13
George V
25
Edward I
35
Henry VII
24
Mary II
6
Edward VIII
1
Edward II
20
Henry VIII
38
Anne
12
George VI
15
Make your own scatterplot. The table shows the ages and weights of ten bears captured by the department of natural resources. 11. Create a scatterplot for the data, using an appropriate scale for the axes. You can use the plot below, or make your own. 12. You should notice the dots fall roughly into a trend going from the lower left to upper right. Does this make sense?
8
Remember: • SHOW ALL WORK. You must communicate to me how your got your solution. This is as important as your solution. • Show your work and your answers on a separate sheet at the end, then you only need to submit that sheet (not these sheets). In the last activity, you found simple probabilities. In this activity, we’ll look at some probabilities that are more complex. We’ll also see how understanding the mathematical concept of sets can help you.
Sets What is a set? In probability, a set is a group of objects with a set of rules to decide if an object belongs with the group or not. “Everyone enrolled in this course” is a set. It’s a group of objects (people), and I can tell who’s in the set (you) and who’s not (President Obama). “Young adults” is not a set unless it’s better defined who is a young adult and who is not. Example 1: Write the elements belonging to each of the following sets. set: natural numbers less than 5 Answer: {1,2,3,4} set: states that border Minnesota Answer: {Iowa, Wisconsin, North Dakota, South Dakota}
What is a subset? A subset is a set completely contained within another set. Example 2: Tell if B is a subset of A. A: natural numbers less than 5 B: {4,5} Answer: no, because 5 is not part of A A: the 50 United States of America B: states that border Minnesota Answer: yes, each of the states that borders Minnesota is a member of the USA
Problem Set 3 1. Answer yes or no if a-f are elements of the set. set: {1,2,3,4,5,6,7,8,9} for example: i. 7 yes
ii. 99 no
a. 2
e. 12
b. 3
c. 10
d. 0
f. 4
2. Answer yes or no if a-e are subsets of the set set: Big Ten schools {Illinois, Indiana, Iowa, Michigan, Michigan State, Minnesota, Northwestern, Ohio State, Penn State, Purdue, Wisconsin} a. {Michigan, Michigan State}
b. {Minnesota}
c. {Iowa, Iowa State}
d. {Indiana State}
e. {Ohio State, Michigan State, Penn State}
Probability What are mutually exclusive events? In probability, mutually exclusive events are two events that can not occur at the same time, such as getting both heads and tails on the same coin flip, or having a student tell you she is in 9th grade and 12th grade at the same time. Example 1: Are the events mutually exclusive? event 1: rolling a single die and getting a 4,5, or 6
event 2: rolling a single die and getting a 1 or 2
yes, these events are mutually exclusive because if you get a 4,5, or 6, that means you didn’t get a 1 or 2 event 1: choosing a random student who is female
event 2: choosing a random student who is in 9th grade
no, these events are not mutually exclusive because you could be a girl in 9th grade
Some probability question ask the probability of event A “OR” event B. To find these probabilities, simply add the probability of A plus the probability of B if A and B are mutually exclusive events. Example 2: What is the probability of drawing from a deck of cards a 2 or a queen. These are mutually exclusive events—a card is never both a 2 and a queen. The probability of drawing a 2 is 4/52. The probability of drawing a queen is also 4/52. 4/52 + 4/52 = 8/52 = 2/13. So the probability is 2/13.
What if the events are not mutually exclusive? You still add the two probabilities of the two separate events, but there’s one more step—subtracting the probability of both. Example 3: If a single card is drawn from a deck of 52, find the probability that it will be a red or a face card. P(red card) = 26 red cards/52 total cards P(face card) = 12 face card/52 total cards P(red) + P(face) = 38/52 P(red and face) = 6 red face cards/52 total cards P(red or face) = 38/52 - 6/52 = 32/52 = 8/13
Problem Set 4 Answer if the events in 1-6 are mutually exclusive (m.e.) or not mutually exclusive 1. owning a car and owning a truck 2. wearing glasses and wearing sandals 3. being married and being over 30 years old 4. being a teenager and being over 30 years old 5. getting a 4 and an odd number on a die you rolled one time 6. being in 12th grade and not being in 12th grade Find the probabilities of drawing the following cards from a standard deck 7. less than a 4 (count aces as ones) 8. a black card or an ace 9. a heart or a jack 10. a red card or a seven
Graphing and Data Analysis What is a bar graph? When we’re keeping track of different categories of data, it can be useful to show how much of the data is in what categories with a bar graph.
8
8
8
Example 1: Each bar represents a number of students who received a bachelor’s degree in different major areas. If twice as many people receive a degree is business as in the social sciences, the business bar will be twice as long as the social sciences bar. One way this could be different from a circle graph is that there might be other disciplines (foreign language, for example) that aren’t shown on this graph. (If you printed this out in black and white, you may want to check out the color coding in the original pdf.)
What is a circle graph? When we’re keeping track of different categories of data, it can be useful to show how much of the data is in what categories with a circle graph, too. It shows proportions, like a bar graph, but in a slightly different way. Example 2: The entire circle represents ALL U.S. public school students, so every student needs to belong in one slice of the pie.
What is a dot plot? A dot plot keeps track of frequency, or how often, different results show up. This is useful when dealing with small amounts of quantitative (numerical) data. Example 3: Each dot represents the population for one of the 50 U.S. states. Minnesota is in red. When different states have the same population, the dot stack up on top of each other.
What is a stemplot? A stemplot is a simple way to visually represent small amounts of quantitative (number) data. This is sometimes called a stem and leaf plot because each piece of data is separated into stem and leaf pieces. Example 4: This table shows student scores on a final exam. Take the first digit in each two digit score, and have it represent the stem. The second digit will be the leaf. In this way, the graph will turn out a little like a dotplot.
student score
A
B
C
D
E
F
G
H
I
J
64 87 92 81 88 73 77 75 80 70
6 4 7 0357 8 0178 9 2
What is a histogram? A histogram is a visual display similar to a stemplot, but it can be used with large data sets. A histogram has two dimension. One is the range of values, divided into equal subunits, and the other is the frequency of data values in those ranges. Example 5: Use the student exam score data above to create a histogram.
What is a scatterplot? A scatterplot compares two quantitative values simultaneously. On a two-dimensional graph, the x-axis axis is labeled with one variable, and the y-axis is labeled with the other. Example 6: Create a scatterplot comparing the variables “mpg” and “weight.” Notice that each dot represents one car.
Problem Set 5 Look at the bar graph to the right. 1. What do the light and dark bars represent? 2. What are the numbers on top of the columns? How do you know? 3. What is the difference between the total math score in 1967 and 1999?
The graph to the right is the same thing as a circle graph, even though it’s not in the shape of a circle. We might call it an area graph. 4. What was the most sold book type in the U.S. according to the graph? 5. Which is more popular, book club books or college books?
Make your own dot plot. In the table, some data is compiled regarding the number of states visited by Sobriety High School students. Add the number of states you’ve visited to the table. student
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
p
number of states visited
5
11
11
14
28
6
6
9
12
18
5
10
4
14
5
8
you
states visited 6. Create a dot plot with the data in the table. You can use this graph here, or re-create it on your own answer sheet. 7. Circle the dot on the dotplot that represents you. 8. What looks to be about the average number of states visited? (you don’t have to be exact, just give a guess that’s close by looking at the dotplot.)
Use the data in the table describing the lengths of reigns of British rulers beginning with William the Conqueror in 1066 to create a stem plot. 9. How long was the longest reign? Who ruled the longest? 10. Create a stemplot from the lengths of reigns. Use the numerals 0-6 on the left of the line. 0 = 00.
ruler
reign
ruler
reign
ruler
reign
ruler
reign
William I
21
Edward III
50
Edward VI
6
George I
13
William II
13
Richard II
22
Mary I
5
George II
33
Henry I
35
Henry IV
13
Elizabeth I
44
George III
59
Stephen
19
Henry V
9
James I
22
George IV
10
Henry II
35
Henry VI
39
Charles I
24
William IV
7
Richard I
10
Edward IV
22
Charles II
25
Victoria
63
John
17
Edward V
0
James II
3
Edward VII
9
Henry III
56
Richard III
2
William III
13
George V
25
Edward I
35
Henry VII
24
Mary II
6
Edward VIII
1
Edward II
20
Henry VIII
38
Anne
12
George VI
15
Make your own scatterplot. The table shows the ages and weights of ten bears captured by the department of natural resources. 11. Create a scatterplot for the data, using an appropriate scale for the axes. You can use the plot below, or make your own. 12. You should notice the dots fall roughly into a trend going from the lower left to upper right. Does this make sense?
