The document discusses probability theory and concepts like sample space, outcomes, events, and the probability of events occurring. It provides examples of calculating probabilities, including rolling dice. The key points are:
- Probability theory is the study of chance using mathematics
- An experiment is a situation with possible results called outcomes
- The sample space includes all possible outcomes
- An event is a subset of outcomes of interest within the sample space
- Probability is calculated as the number of favorable outcomes divided by the total number of possible outcomes
- Examples are provided to demonstrate calculating probabilities, such as the probability of rolling a 7 when rolling two fair dice.
This is my slide deck from my session at the North Carolina Reading Conference last week in Raleigh, NC. I do staff development to schools and districts all over the country about best practices in literacy instruction. This topic is one of my most requested.
Probability is the way of expressing knowledge of belief that an event will occur on chance.
Did You Know? Probability originated from the Latin word meaning approval.
Make use of the PPT to have a better understanding of Probability.
This is my slide deck from my session at the North Carolina Reading Conference last week in Raleigh, NC. I do staff development to schools and districts all over the country about best practices in literacy instruction. This topic is one of my most requested.
Probability is the way of expressing knowledge of belief that an event will occur on chance.
Did You Know? Probability originated from the Latin word meaning approval.
Make use of the PPT to have a better understanding of Probability.
A random variable is a mathematical concept used to describe the outcome of a random process or experiment. It is a variable that takes on different values based on the outcome of a random event. In mathematical terms, a random variable is a function that maps the outcomes of a random experiment to a set of real numbers.
There are two types of random variables: discrete and continuous. A discrete random variable can take on only a finite or countably infinite number of values, such as the number of heads in a sequence of coin flips. A continuous random variable, on the other hand, can take on any value within a given range, such as the height of a person.
The probability distribution of a random variable defines the likelihood of each possible outcome of the random process or experiment. For a discrete random variable, this is represented by a probability mass function (PMF), which gives the probability of each possible outcome. For a continuous random variable, the probability distribution is represented by a probability density function (PDF), which gives the relative likelihood of different outcomes within a given range.
Another important concept in the study of random variables is expected value, also known as the mean or average of a random variable. The expected value represents the long-term average of the values that a random variable takes on, and is calculated as the weighted sum of the possible values, where the weights are the probabilities of each outcome.
Random variables are used in a variety of mathematical and statistical applications, including decision theory, finance, and quality control. They also play a central role in the study of probability and statistics, and are used to model and analyze complex systems and phenomena in fields such as physics, engineering, and economics.
In conclusion, random variables are a powerful tool for describing the outcomes of random processes and experiments, and provide a framework for understanding the probabilistic behavior of complex systems. Understanding and using random variables is essential for making informed decisions and solving problems in a variety of fields and applications.A random variable is a mathematical concept that describes the outcome of a random event. It is a variable that can take on different values based on the outcome of the event. There are two types of random variables: discrete and continuous. A discrete random variable can take on only a finite or countably infinite number of values, such as the number of heads in a sequence of coin flips. A continuous random variable, on the other hand, can take on any value within a given range, such as the height of a person. The probability distribution of a random variable defines the likelihood of each possible outcome and is used to calculate the expected value, which is the long-term average of the values that a random variable takes on. Random variables are widely used in mathematics and statistics, and are essen
A random variable is a mathematical concept used to describe the outcome of a random process or experiment. It is a variable that takes on different values based on the outcome of a random event. In mathematical terms, a random variable is a function that maps the outcomes of a random experiment to a set of real numbers.
There are two types of random variables: discrete and continuous. A discrete random variable can take on only a finite or countably infinite number of values, such as the number of heads in a sequence of coin flips. A continuous random variable, on the other hand, can take on any value within a given range, such as the height of a person.
The probability distribution of a random variable defines the likelihood of each possible outcome of the random process or experiment. For a discrete random variable, this is represented by a probability mass function (PMF), which gives the probability of each possible outcome. For a continuous random variable, the probability distribution is represented by a probability density function (PDF), which gives the relative likelihood of different outcomes within a given range.
Another important concept in the study of random variables is expected value, also known as the mean or average of a random variable. The expected value represents the long-term average of the values that a random variable takes on, and is calculated as the weighted sum of the possible values, where the weights are the probabilities of each outcome.
Random variables are used in a variety of mathematical and statistical applications, including decision theory, finance, and quality control. They also play a central role in the study of probability and statistics, and are used to model and analyze complex systems and phenomena in fields such as physics, engineering, and economics.
In conclusion, random variables are a powerful tool for describing the outcomes of random processes and experiments, and provide a framework for understanding the probabilistic behavior of complex systems. Understanding and using random variables is essential for making informed decisions and solving problems in a variety of fields and applications.A random variable is a mathematical concept that describes the outcome of a random event. It is a variable that can take on different values based on the outcome of the event. There are two types of random variables: discrete and continuous. A discrete random variable can take on only a finite or countably infinite number of values, such as the number of heads in a sequence of coin flips. A continuous random variable, on the other hand, can take on any value within a given range, such as the height of a person. The probability distribution of a random variable defines the likelihood of each possible outcome and is used to calculate the expected value, which is the long-term average of the values that a random variable takes on. Random variables are widely used in mathematics and statistics, and are essen
Connector Corner: Automate dynamic content and events by pushing a buttonDianaGray10
Here is something new! In our next Connector Corner webinar, we will demonstrate how you can use a single workflow to:
Create a campaign using Mailchimp with merge tags/fields
Send an interactive Slack channel message (using buttons)
Have the message received by managers and peers along with a test email for review
But there’s more:
In a second workflow supporting the same use case, you’ll see:
Your campaign sent to target colleagues for approval
If the “Approve” button is clicked, a Jira/Zendesk ticket is created for the marketing design team
But—if the “Reject” button is pushed, colleagues will be alerted via Slack message
Join us to learn more about this new, human-in-the-loop capability, brought to you by Integration Service connectors.
And...
Speakers:
Akshay Agnihotri, Product Manager
Charlie Greenberg, Host
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Welcome to UiPath Test Automation using UiPath Test Suite series part 3. In this session, we will cover desktop automation along with UI automation.
Topics covered:
UI automation Introduction,
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Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
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We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
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Neuro-symbolic (NeSy) AI is on the rise. However, simply machine learning on just any symbolic structure is not sufficient to really harvest the gains of NeSy. These will only be gained when the symbolic structures have an actual semantics. I give an operational definition of semantics as “predictable inference”.
All of this illustrated with link prediction over knowledge graphs, but the argument is general.
Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
💥 Speed, accuracy, and scaling – discover the superpowers of GenAI in action with UiPath Document Understanding and Communications Mining™:
See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
This is a hands-on session specifically designed for automation developers and AI enthusiasts seeking to enhance their knowledge in leveraging the latest intelligent document processing capabilities offered by UiPath.
Speakers:
👨🏫 Andras Palfi, Senior Product Manager, UiPath
👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
Key Trends Shaping the Future of Infrastructure.pdfCheryl Hung
Keynote at DIGIT West Expo, Glasgow on 29 May 2024.
Cheryl Hung, ochery.com
Sr Director, Infrastructure Ecosystem, Arm.
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• Test Automation: How AI-powered test case generation, optimization, and self-healing tests are making testing more efficient and effective.
• Visual Testing: Explore the emerging capabilities of AI in visual testing and how it's set to revolutionize UI verification.
• Inflectra's AI Solutions: See demonstrations of Inflectra's cutting-edge AI tools like the ChatGPT plugin and Azure Open AI platform, designed to streamline your testing process.
Whether you're a developer, tester, or QA professional, this webinar will give you valuable insights into how AI is shaping the future of software delivery.
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
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The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
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Bob Boule
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Gopinath Rebala
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"Impact of front-end architecture on development cost", Viktor TurskyiFwdays
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Search and Society: Reimagining Information Access for Radical FuturesBhaskar Mitra
The field of Information retrieval (IR) is currently undergoing a transformative shift, at least partly due to the emerging applications of generative AI to information access. In this talk, we will deliberate on the sociotechnical implications of generative AI for information access. We will argue that there is both a critical necessity and an exciting opportunity for the IR community to re-center our research agendas on societal needs while dismantling the artificial separation between the work on fairness, accountability, transparency, and ethics in IR and the rest of IR research. Instead of adopting a reactionary strategy of trying to mitigate potential social harms from emerging technologies, the community should aim to proactively set the research agenda for the kinds of systems we should build inspired by diverse explicitly stated sociotechnical imaginaries. The sociotechnical imaginaries that underpin the design and development of information access technologies needs to be explicitly articulated, and we need to develop theories of change in context of these diverse perspectives. Our guiding future imaginaries must be informed by other academic fields, such as democratic theory and critical theory, and should be co-developed with social science scholars, legal scholars, civil rights and social justice activists, and artists, among others.
2. Warm-up
A drawer contains r red socks, b blue socks, and w white socks.
Assume that you draw a sock at random from the drawer.
1. What is the probability that the sock is red?
2. What is the probability that the sock is not white?
3. What is the probability that the sock is green?
3. Warm-up
A drawer contains r red socks, b blue socks, and w white socks.
Assume that you draw a sock at random from the drawer.
1. What is the probability that the sock is red?
r
r+b+w
2. What is the probability that the sock is not white?
3. What is the probability that the sock is green?
4. Warm-up
A drawer contains r red socks, b blue socks, and w white socks.
Assume that you draw a sock at random from the drawer.
1. What is the probability that the sock is red?
r
r+b+w
2. What is the probability that the sock is not white?
r+b
r+b+w
3. What is the probability that the sock is green?
5. Warm-up
A drawer contains r red socks, b blue socks, and w white socks.
Assume that you draw a sock at random from the drawer.
1. What is the probability that the sock is red?
r
r+b+w
2. What is the probability that the sock is not white?
r+b
r+b+w
3. What is the probability that the sock is green?
0
9. Probability theory: The study of chance using math
Experiment: A situation with several possible results
10. Probability theory: The study of chance using math
Experiment: A situation with several possible results
Outcome:
11. Probability theory: The study of chance using math
Experiment: A situation with several possible results
Outcome: A possible result of an experiment
12. Probability theory: The study of chance using math
Experiment: A situation with several possible results
Outcome: A possible result of an experiment
Sample Space:
13. Probability theory: The study of chance using math
Experiment: A situation with several possible results
Outcome: A possible result of an experiment
Sample Space: The set of the total number of possible outcomes
of an experiment
14. Example 1
A standard deck of playing cards is used to draw a card at random.
a. Give the sample space and describe some of the
characteristics of the deck.
15. Example 1
A standard deck of playing cards is used to draw a card at random.
a. Give the sample space and describe some of the
characteristics of the deck.
The sample space is 52, since there are 52 cards.
16. Example 1
A standard deck of playing cards is used to draw a card at random.
a. Give the sample space and describe some of the
characteristics of the deck.
The sample space is 52, since there are 52 cards.
There are 26 red (13 hearts, 13 diamonds) and 26 black (13 clubs,
13 spades)
17. Example 1
A standard deck of playing cards is used to draw a card at random.
a. Give the sample space and describe some of the
characteristics of the deck.
The sample space is 52, since there are 52 cards.
There are 26 red (13 hearts, 13 diamonds) and 26 black (13 clubs,
13 spades)
Each suit has 4 face cards (J, Q, K, A) for a total of 16 in the deck
18. Example 1
A standard deck of playing cards is used to draw a card at random.
a. Give the sample space and describe some of the
characteristics of the deck.
The sample space is 52, since there are 52 cards.
There are 26 red (13 hearts, 13 diamonds) and 26 black (13 clubs,
13 spades)
Each suit has 4 face cards (J, Q, K, A) for a total of 16 in the deck
Anything else?
24. Event: Any subset of the sample space or experiment
This means that it is a particular occurrence of something
happening
25. Event: Any subset of the sample space or experiment
This means that it is a particular occurrence of something
happening
Probability that E occurs:
26. Event: Any subset of the sample space or experiment
This means that it is a particular occurrence of something
happening
Probability that E occurs: The probability that the given event
will occur
27. Event: Any subset of the sample space or experiment
This means that it is a particular occurrence of something
happening
Probability that E occurs: The probability that the given event
will occur
So, E is an event within our sample space S. To do this, we
need to know the size of our sample space and the size of
our favorable outcomes.
28. Event: Any subset of the sample space or experiment
This means that it is a particular occurrence of something
happening
Probability that E occurs: The probability that the given event
will occur
So, E is an event within our sample space S. To do this, we
need to know the size of our sample space and the size of
our favorable outcomes.
# of favorable outcomes
P(E) =
# of possible outcomes in S
32. Fair: When each outcome is as likely as another
Biased: When all outcomes are not equally likely to occur
33. Example 2
Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
34. Example 2
Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space:
35. Example 2
Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1
36. Example 2
Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2
37. Example 2
Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3
38. Example 2
Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4
39. Example 2
Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5
40. Example 2
Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5 1,6
41. Example 2
Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5 1,6
2,1
42. Example 2
Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5 1,6
2,1 2,2
43. Example 2
Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5 1,6
2,1 2,2 2,3
44. Example 2
Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5 1,6
2,1 2,2 2,3 2,4
45. Example 2
Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5 1,6
2,1 2,2 2,3 2,4 2,5
46. Example 2
Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5 1,6
2,1 2,2 2,3 2,4 2,5 2,6
47. Example 2
Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 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
48. Example 2
Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 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
49. Example 2
Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 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
Highest Sum:
50. Example 2
Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 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
Highest Sum: 7
51. Example 2
Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 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
Highest Sum: 7
P(7) =
52. Example 2
Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 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
Highest Sum: 7
# of sum of 7
P(7) =
# of possible sums
53. Example 2
Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 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
Highest Sum: 7
# of sum of 7 6
P(7) = =
# of possible sums 36
54. Example 2
Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 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
Highest Sum: 7
# of sum of 7 6 1
P(7) = = =
# of possible sums 36 6
55. Example 2
Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 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
Highest Sum: 7
# of sum of 7 6 1
P(7) = = = = 16 2/3% chance of
# of possible sums 36 6 rolling a 7
56. Example 3
Persons A, B, and C were playing a game and needed to deicide
who would go first. Person A suggested tossing two dice. If a 3,
6, 9, or 12 appeared, person A would go first. If a 2, 5, 8, or 11
appeared, person B would go first. If a 1, 4, 7, or 10 appeared,
person C would go first. Is this fair? Why or why not?
57. Example 3
Persons A, B, and C were playing a game and needed to deicide
who would go first. Person A suggested tossing two dice. If a 3,
6, 9, or 12 appeared, person A would go first. If a 2, 5, 8, or 11
appeared, person B would go first. If a 1, 4, 7, or 10 appeared,
person C would go first. Is this fair? Why or why not?
P(3, 6, 9, or 12)
58. Example 3
Persons A, B, and C were playing a game and needed to deicide
who would go first. Person A suggested tossing two dice. If a 3,
6, 9, or 12 appeared, person A would go first. If a 2, 5, 8, or 11
appeared, person B would go first. If a 1, 4, 7, or 10 appeared,
person C would go first. Is this fair? Why or why not?
12
P(3, 6, 9, or 12) =
36
59. Example 3
Persons A, B, and C were playing a game and needed to deicide
who would go first. Person A suggested tossing two dice. If a 3,
6, 9, or 12 appeared, person A would go first. If a 2, 5, 8, or 11
appeared, person B would go first. If a 1, 4, 7, or 10 appeared,
person C would go first. Is this fair? Why or why not?
12 1
P(3, 6, 9, or 12) = =
36 3
60. Example 3
Persons A, B, and C were playing a game and needed to deicide
who would go first. Person A suggested tossing two dice. If a 3,
6, 9, or 12 appeared, person A would go first. If a 2, 5, 8, or 11
appeared, person B would go first. If a 1, 4, 7, or 10 appeared,
person C would go first. Is this fair? Why or why not?
12 1
P(3, 6, 9, or 12) = =
36 3
P(2, 5, 8, or 11)
61. Example 3
Persons A, B, and C were playing a game and needed to deicide
who would go first. Person A suggested tossing two dice. If a 3,
6, 9, or 12 appeared, person A would go first. If a 2, 5, 8, or 11
appeared, person B would go first. If a 1, 4, 7, or 10 appeared,
person C would go first. Is this fair? Why or why not?
12 1
P(3, 6, 9, or 12) = =
36 3
12 1
P(2, 5, 8, or 11) = =
36 3
62. Example 3
Persons A, B, and C were playing a game and needed to deicide
who would go first. Person A suggested tossing two dice. If a 3,
6, 9, or 12 appeared, person A would go first. If a 2, 5, 8, or 11
appeared, person B would go first. If a 1, 4, 7, or 10 appeared,
person C would go first. Is this fair? Why or why not?
12 1
P(3, 6, 9, or 12) = =
36 3
12 1
P(2, 5, 8, or 11) = =
36 3
12 1
P(1, 4, 7, or 10) = =
36 3
63. Example 3
Persons A, B, and C were playing a game and needed to deicide
who would go first. Person A suggested tossing two dice. If a 3,
6, 9, or 12 appeared, person A would go first. If a 2, 5, 8, or 11
appeared, person B would go first. If a 1, 4, 7, or 10 appeared,
person C would go first. Is this fair? Why or why not?
12 1
P(3, 6, 9, or 12) = =
36 3 This is fair
12 1 since all of the
P(2, 5, 8, or 11) = =
36 3 probabilities
are the same
12 1
P(1, 4, 7, or 10) = =
36 3
65. Theorem: Basic Properties
of Probability
Let S be the sample space associated with an experiment and
P(E) be the probability of E.
66. Theorem: Basic Properties
of Probability
Let S be the sample space associated with an experiment and
P(E) be the probability of E.
0 ≤ P(E) ≤ 1
67. Theorem: Basic Properties
of Probability
Let S be the sample space associated with an experiment and
P(E) be the probability of E.
0 ≤ P(E) ≤ 1
If E = S, then P(E) = 1
68. Theorem: Basic Properties
of Probability
Let S be the sample space associated with an experiment and
P(E) be the probability of E.
0 ≤ P(E) ≤ 1
If E = S, then P(E) = 1
If E = ∅, then P(E) = 0
69. Example 4
If births of boys and girls are assumed equally likely, what is the
probability that a family with four children has all girls?
70. Example 4
If births of boys and girls are assumed equally likely, what is the
probability that a family with four children has all girls?
Make a list of all possible outcomes
BBBB BBBG BBGB BGBB
BBGG BGGB GGBB BGBG
GBGB BGGB GBGG BGGG
GGBG GGGB GBBB GGGG
71. Example 4
If births of boys and girls are assumed equally likely, what is the
probability that a family with four children has all girls?
Make a list of all possible outcomes
BBBB BBBG BBGB BGBB
BBGG BGGB GGBB BGBG
GBGB BGGB GBGG BGGG
GGBG GGGB GBBB GGGG
One situation out of the sixteen gives 4 girls, so 1/16, or 6.25%
