Casino 1
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Casino Games Activity 1
8
At some point in your school career, you’ve probably learned something about probability and statistics. Our goal in this activity is to review stuff you’ve probably learned already and refresh our memories so that we can learn new stuff during the rest of the course. This should be a fairly easy activity to fill out. Some guidelines: • 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).
Basic Probability What is a sample space? In probability, an experiment is an activity or occurrence with an observable result. Flipping a coin, rolling a die, and drawing a name are all experiments. Each repetition of an experiment is called a trial. The possible results of each trial are called outcomes. The set of all possible outcomes for an experiment is the sample space for that experiment. Example 1: You are flipping a coin. Question
Answer
What is the experiment?
tossing a coin
What is one trial in this experiment?
one coin flip
What are the possible outcomes?
the coin could land heads or tails
What is the sample space?
The sample space is all the possible outcomes: sample space = {heads, tails}
Example 2: Someone is studying the the numbers of boys and girls in families with exactly 3 kids. Question
Answer
What is the experiment?
finding out numbers of boys (b) and girls (g) in a 3 kid family
What is one trial in this experiment?
recording the boys and girls in one 3 kid family
What are the possible outcomes?
bbb, ggg, bbg, ggb (assuming order doesn’t matter)
What is the sample space?
The sample space is all the possible outcomes: sample space = {bbb, ggg, bbg, ggb}
What is an event? In probability, an event is a thing in the sample space. Flipping heads on a coin is an event. A family having all girls (ggg) is an event. Example 3: Your friend is rolling a die. Question
Answer
What is the experiment?
rolling the die
What is one trial in this experiment?
one roll of the die
What are the possible outcomes?
you could roll a 1, 2, 3, 4, 5, or 6
What is the sample space?
With a 6-sided die, the sample space is {1, 2, 3, 4, 5, 6}.
What is an example of an event?
“The die shows a 4” is an example of an event.
These are two dice. Taken one at a time, the word is die.
How do you calculate basic probabilities? When the outcomes in the sample space are all equally likely, the probability of an event is figured out by dividing the number of ways the event can occur by the number of things in the sample space. Example 4: What’s the probability the die will show an even number? There are 3 ways this event can occur: {2, 4, 6} There are 6 things in the sample space: {1, 2, 3, 4, 5, 6} Therefore, the probability the die will show an even number is 3/6, which is the same is 1/2. Example 5: What’s the probability the die will show a number less than 10? There are 6 ways this event can occur: {1, 2, 3, 4, 5, 6} There are 6 things in the sample space: {1, 2, 3, 4, 5, 6} Therefore, the probability the die will show a number less than 10 is 6/6, which is the same is 1. Example 6: What’s the probability the die will show an 8? There are 8 ways this event can occur. There are 6 things in the sample space: {1, 2, 3, 4, 5, 6} Therefore, the probability the die will show an 8 is 0/6, which is the same is 0.
Problem set 1: Write the sample space for the following experiments: 1. A month of the year is chosen for a wedding 2. A student is asked how many points she earned on a recent 80-point test. 3. A person is asked the number of hours (to the nearest hour) he watched television yesterday. E1. A coin is tossed, and a die is rolled. As you probably know, a deck of cards has 52 cards in four suits: hearts, spades, clubs, and diamonds. Each suit has 13 cards. 4. What is the probability of drawing an ace? 5. What is the probability of drawing a face card (a king, queen, or jack)? 6. What is the probability of drawing a spade? 7. What is the probability of drawing a black 9? E2. What is the probability of drawing a spade or a heart? 8. The student sitting next to you concludes that the probability of the ceiling falling down on both of you before class ends is 1/2, because there are two possible outcomes—the ceiling will fall or not fall. What is wrong with this reasoning?
Basic Statistics What is the mean? You may know already that there are different averages. When most people talk about “finding the average,” they are talking about the mean. The mean is calculated by adding up a set of numbers, then dividing by the number of numbers. Example 1: Five students tell the number of days they’ve been clean or sober.
student
Lindsay
Swathi
Michael
Chris
Sarah
days
170
430
30
5
100
Add the data: 170 + 430 + 30 + 5 + 100 = 735 There are 5 data, so divide the total by 5: 735/5 = 147 The mean number of days sober these students have is 147.
What is the median? A different average is the median. The median is the middle number, and that’s easy to remember because median and middle both start with m—vowel—d. To find a median, arrange the data in order from lowest to highest (or highest to lowest), then the median is the middle number. If there is an even number of entries, the median is the mean of the two center entries. Example 2: Five students tell the number of days they’ve been clean or sober.
student
Lindsay
Swathi
Michael
Chris
Sarah
days
170
430
30
5
100
Put the data in order: 5, 30, 100, 170, 430 100 is the middle number, so 100 is the median. Example 3: Another student joins the group.
student
Lindsay
Swathi
Michael
Chris
Sarah
Katy
days
170
430
30
5
100
900
Put the data in order: 5, 30, 100, 170, 430, 900 100 and 170 are the middle numbers, so find their mean: 100 + 170 = 270. 270/2 = 135. 135 is the median.
What is the mode? A different average is the mode. Whatever number(s) occurs the most often is the mode. Example 4: Five other students compare their time.
student
Nick
Janaya
Brittany
John
Neeko
days
35
42
35
97
44
Two students have 35 days, so 35 is the mode. Example 5: Another student joins the group.
student
Nick
Janaya
Brittany
John
Michael
Neeko
days
35
42
35
97
44
44
Two students have 35 days, and two have 44 days, so 35 and 44 are both modes.
8
8
8
What is the range? While an average gives us a sense of where most of the data is, it doesn’t tell us about how much the different values vary from each other. The range is one measure that does do this. To find the range, take the highest number minus the lowest. Example 6: These same six students tell the number of days they’ve been clean or sober.
student
Nick
Janaya
Brittany
John
Michael
Neeko
days
35
42
35
97
44
44
The range of the data is 97 (highest) - 35 (lowest) = 62. 62 is the range.
What is the standard deviation? The range only takes into account two values. The standard deviation is calculated with a much more complex process that is affected by every single data value. The process is too complex to make it worthwhile for us to calculate this by hand. We’ll use a calculator instead. On the internet, type “standard deviation calculator” and use that to find standard deviations. Problem set 2: If you’re using a calculator to find these statistical values, say so. Then instead of showing work, describe what you did. Use the roller coaster data on the following page to answer these questions. 1. What is the mean value for “largest drop?” 2. What is the mean value for “top speed?” 3. What is the median value for “largest drop?” 4. What is the median value for “top speed?” 5. What is the mode value for “largest drop?” 6. What is the mode value for “top speed?” 7. What is the range for “largest drop?” 8. What is the range for “top speed?” 9. What is the standard deviation for “largest drop?” 10. What is the standard deviation for “top speed?”
8
At some point in your school career, you’ve probably learned something about probability and statistics. Our goal in this activity is to review stuff you’ve probably learned already and refresh our memories so that we can learn new stuff during the rest of the course. This should be a fairly easy activity to fill out. Some guidelines: • 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).
Basic Probability What is a sample space? In probability, an experiment is an activity or occurrence with an observable result. Flipping a coin, rolling a die, and drawing a name are all experiments. Each repetition of an experiment is called a trial. The possible results of each trial are called outcomes. The set of all possible outcomes for an experiment is the sample space for that experiment. Example 1: You are flipping a coin. Question
Answer
What is the experiment?
tossing a coin
What is one trial in this experiment?
one coin flip
What are the possible outcomes?
the coin could land heads or tails
What is the sample space?
The sample space is all the possible outcomes: sample space = {heads, tails}
Example 2: Someone is studying the the numbers of boys and girls in families with exactly 3 kids. Question
Answer
What is the experiment?
finding out numbers of boys (b) and girls (g) in a 3 kid family
What is one trial in this experiment?
recording the boys and girls in one 3 kid family
What are the possible outcomes?
bbb, ggg, bbg, ggb (assuming order doesn’t matter)
What is the sample space?
The sample space is all the possible outcomes: sample space = {bbb, ggg, bbg, ggb}
What is an event? In probability, an event is a thing in the sample space. Flipping heads on a coin is an event. A family having all girls (ggg) is an event. Example 3: Your friend is rolling a die. Question
Answer
What is the experiment?
rolling the die
What is one trial in this experiment?
one roll of the die
What are the possible outcomes?
you could roll a 1, 2, 3, 4, 5, or 6
What is the sample space?
With a 6-sided die, the sample space is {1, 2, 3, 4, 5, 6}.
What is an example of an event?
“The die shows a 4” is an example of an event.
These are two dice. Taken one at a time, the word is die.
How do you calculate basic probabilities? When the outcomes in the sample space are all equally likely, the probability of an event is figured out by dividing the number of ways the event can occur by the number of things in the sample space. Example 4: What’s the probability the die will show an even number? There are 3 ways this event can occur: {2, 4, 6} There are 6 things in the sample space: {1, 2, 3, 4, 5, 6} Therefore, the probability the die will show an even number is 3/6, which is the same is 1/2. Example 5: What’s the probability the die will show a number less than 10? There are 6 ways this event can occur: {1, 2, 3, 4, 5, 6} There are 6 things in the sample space: {1, 2, 3, 4, 5, 6} Therefore, the probability the die will show a number less than 10 is 6/6, which is the same is 1. Example 6: What’s the probability the die will show an 8? There are 8 ways this event can occur. There are 6 things in the sample space: {1, 2, 3, 4, 5, 6} Therefore, the probability the die will show an 8 is 0/6, which is the same is 0.
Problem set 1: Write the sample space for the following experiments: 1. A month of the year is chosen for a wedding 2. A student is asked how many points she earned on a recent 80-point test. 3. A person is asked the number of hours (to the nearest hour) he watched television yesterday. E1. A coin is tossed, and a die is rolled. As you probably know, a deck of cards has 52 cards in four suits: hearts, spades, clubs, and diamonds. Each suit has 13 cards. 4. What is the probability of drawing an ace? 5. What is the probability of drawing a face card (a king, queen, or jack)? 6. What is the probability of drawing a spade? 7. What is the probability of drawing a black 9? E2. What is the probability of drawing a spade or a heart? 8. The student sitting next to you concludes that the probability of the ceiling falling down on both of you before class ends is 1/2, because there are two possible outcomes—the ceiling will fall or not fall. What is wrong with this reasoning?
Basic Statistics What is the mean? You may know already that there are different averages. When most people talk about “finding the average,” they are talking about the mean. The mean is calculated by adding up a set of numbers, then dividing by the number of numbers. Example 1: Five students tell the number of days they’ve been clean or sober.
student
Lindsay
Swathi
Michael
Chris
Sarah
days
170
430
30
5
100
Add the data: 170 + 430 + 30 + 5 + 100 = 735 There are 5 data, so divide the total by 5: 735/5 = 147 The mean number of days sober these students have is 147.
What is the median? A different average is the median. The median is the middle number, and that’s easy to remember because median and middle both start with m—vowel—d. To find a median, arrange the data in order from lowest to highest (or highest to lowest), then the median is the middle number. If there is an even number of entries, the median is the mean of the two center entries. Example 2: Five students tell the number of days they’ve been clean or sober.
student
Lindsay
Swathi
Michael
Chris
Sarah
days
170
430
30
5
100
Put the data in order: 5, 30, 100, 170, 430 100 is the middle number, so 100 is the median. Example 3: Another student joins the group.
student
Lindsay
Swathi
Michael
Chris
Sarah
Katy
days
170
430
30
5
100
900
Put the data in order: 5, 30, 100, 170, 430, 900 100 and 170 are the middle numbers, so find their mean: 100 + 170 = 270. 270/2 = 135. 135 is the median.
What is the mode? A different average is the mode. Whatever number(s) occurs the most often is the mode. Example 4: Five other students compare their time.
student
Nick
Janaya
Brittany
John
Neeko
days
35
42
35
97
44
Two students have 35 days, so 35 is the mode. Example 5: Another student joins the group.
student
Nick
Janaya
Brittany
John
Michael
Neeko
days
35
42
35
97
44
44
Two students have 35 days, and two have 44 days, so 35 and 44 are both modes.
8
8
8
What is the range? While an average gives us a sense of where most of the data is, it doesn’t tell us about how much the different values vary from each other. The range is one measure that does do this. To find the range, take the highest number minus the lowest. Example 6: These same six students tell the number of days they’ve been clean or sober.
student
Nick
Janaya
Brittany
John
Michael
Neeko
days
35
42
35
97
44
44
The range of the data is 97 (highest) - 35 (lowest) = 62. 62 is the range.
What is the standard deviation? The range only takes into account two values. The standard deviation is calculated with a much more complex process that is affected by every single data value. The process is too complex to make it worthwhile for us to calculate this by hand. We’ll use a calculator instead. On the internet, type “standard deviation calculator” and use that to find standard deviations. Problem set 2: If you’re using a calculator to find these statistical values, say so. Then instead of showing work, describe what you did. Use the roller coaster data on the following page to answer these questions. 1. What is the mean value for “largest drop?” 2. What is the mean value for “top speed?” 3. What is the median value for “largest drop?” 4. What is the median value for “top speed?” 5. What is the mode value for “largest drop?” 6. What is the mode value for “top speed?” 7. What is the range for “largest drop?” 8. What is the range for “top speed?” 9. What is the standard deviation for “largest drop?” 10. What is the standard deviation for “top speed?”
