Sal shows examples of intersection and union of sets and introduces some set notation.

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Introduction to Probability is a resource that provides assessment and teacher tutorials that focus on mathematic concepts such as probability, outcomes, events, probability distribution, laws, values and odd.

The microscopic world is full of phenomena very different from what we see in everyday life. Some of those phenomena can only be explained using quantum mechanics. This activity introduces basic quantum mechanics concepts about electrons that are essential to understanding modern and future technology, especially nanotechnology. Start by exploring probability distribution, then discover the behavior of electrons with a series of simulations.

Students learn about complex networks and how to use graphs to represent them. They also learn that graph theory is a useful part of mathematics for studying complex networks in diverse applications of science and engineering, including neural networks in the brain, biochemical reaction networks in cells, communication networks, such as the internet, and social networks. Students are also introduced to random processes on networks. An illustrative example shows how a random process can be used to represent the spread of an infectious disease, such as the flu, on a social network of students, and demonstrates how scientists and engineers use mathematics and computers to model and simulate random processes on complex networks for the purposes of learning more about our world and creating solutions to improve our health, happiness and safety.

This model-eliciting activity has students create rules to allow them to judge whether or not the shuffle feature on a particular iPod appears to produce randomly generated playlists. Because people's intuitions about random events and randomly generated data are often incorrect or misleading, this activity initially focuses students' attention on describing characteristics of 25 playlists that were randomly generated. Students then use these characteristics to come up with rules for judging whether a playlist does NOT appear to be randomly generated. Students test and revise their rules (model) using five additional playlsits. Then, they apply their model to three particular playlists that have been submitted to Apple by an unhappy iPod owner who claims the shuffle feature on his iPod is not generating random playlists. In the final part of the activity, students write a letter to the ipod owner, on behalf of Apple, explaining the use of their model and their final conclusion about whether these three suspicious playlists appear to have been randomly generated.This lesson provides an introduction to the fundamental ideas of randomness, random sequences and random samples.

In this video, explore the odds of winning the Powerball lottery and learn what few people realize about their chances of winning: they are miniscule. In the accompanying classroom activity, students create simple lottery models to help them understand the probability of compound events. Though the probability of success at these experiments is better than that of Powerball, students will quickly see that the odds of winning any lottery-type game are seldom in their favor.

Discusses how to make a reasonable prediction of how many times a spinner will land on a given symbol when you spin a few hundred times. [5:05]

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Practice making predictions with probability with these problems.

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Samples and ProbabilityType of Unit: ConceptualPrior KnowledgeStudents should be able to:Understand the concept of a ratio.Write ratios as percents.Describe data using measures of center.Display and interpret data in dot plots, histograms, and box plots.Lesson FlowStudents begin to think about probability by considering the relative likelihood of familiar events on the continuum between impossible and certain. Students begin to formalize this understanding of probability. They are introduced to the concept of probability as a measure of likelihood, and how to calculate probability of equally likely events using a ratio. The terms (impossible, certain, etc.) are given numerical values. Next, students compare expected results to actual results by calculating the probability of an event and conducting an experiment. Students explore the probability of outcomes that are not equally likely. They collect data to estimate the experimental probabilities. They use ratio and proportion to predict results for a large number of trials. Students learn about compound events. They use tree diagrams, tables, and systematic lists as tools to find the sample space. They determine the theoretical probability of first independent, and then dependent events. In Lesson 10 students identify a question to investigate for a unit project and submit a proposal. They then complete a Self Check. In Lesson 11, students review the results of the Self Check, solve a related problem, and take a Quiz.Students are introduced to the concept of sampling as a method of determining characteristics of a population. They consider how a sample can be random or biased, and think about methods for randomly sampling a population to ensure that it is representative. In Lesson 13, students collect and analyze data for their unit project. Students begin to apply their knowledge of statistics learned in sixth grade. They determine the typical class score from a sample of the population, and reason about the representativeness of the sample. Then, students begin to develop intuition about appropriate sample size by conducting an experiment. They compare different sample sizes, and decide whether increasing the sample size improves the results. In Lesson 16 and Lesson 17, students compare two data sets using any tools they wish. Students will be reminded of Mean Average Deviation (MAD), which will be a useful tool in this situation. Students complete another Self Check, review the results of their Self Check, and solve additional problems. The unit ends with three days for students to work on Gallery problems, possibly using one of the days to complete their project or get help on their project if needed, two days for students to present their unit projects to the class, and one day for the End of Unit Assessment.

Students begin to formalize their understanding of probability. They are introduced to the concept of probability as a measure of likelihood and how to calculate probability as a ratio. The terms discussed (impossible, certain, etc.) in Lesson 1 are given numerical values.Key ConceptsStudents will think of probability as a ratio; it can be written as a fraction, decimal, or a percent ranging from 0 to 1.Students will think about ratio and proportion to predict results.Goals and Learning ObjectivesDefine probability as a measure of likelihood and the ratio of favorable outcomes to the total number of outcomes for an event.Predict results based on theoretical probability using ratio and proportion.

Students extend their understanding of compound events. They will compare experimental results to predicted results by calculating the probability of an event, then conducting an experiment.Key ConceptsStudents apply their understanding of compound events to actual experiments.Students will see there is variability in actual results.Goals and Learning ObjectivesContinue to explore compound independent events.Compare theoretical probability to experimental probability.

Students begin learning about compound events by considering independent events. They will consider everyday objects with known probabilities. Students will represent sample spaces using lists, tables, and tree diagrams in order to calculate the probability of certain events.Key ConceptsCompound events are introduced in this lesson, building upon what students have learned about determining sample space and probabilities of single events.Terms introduced are:multistage experiment: an experiment in which more than one action is performedcompound events: the combined results of multistage experimentsindependent events: compound events in which the outcome of one does not affect the outcome of the otherGoals and Learning ObjectivesLearn about compound events and sample spaces.Use different tools to find the sample space (tree diagrams, tables, lists) of a compound event.Use ratio and proportion to solve problems.SWD: Go over the mathematical language used throughout the module. Make sure students use that language when discussing problems in this lesson.

Students will continue to apply their understanding of compound independent events. They will calculate probabilities and represent sample spaces with visual representations.Key ConceptsStudents continue to solve problems with compound events. The formula for calculating the probability of independent events is introduced:P(A and B) = P(A) ⋅ P(B)Goals and Learning ObjectivesDeepen understanding of compound events using lists, tables, and tree diagrams.Learn about the Fundamental Counting Principle.

Gallery OverviewAllow students who have a clear understanding of the content thus far in the unit to work on Gallery problems of their choosing. You can then use this time to provide additional help to students who need review of the unit's concepts or to assist students who may have fallen behind on work.Chance of RainStudents are given the probability that it will rain on two different days and asked to find the chance that it will rain on one of the two days.PenguinsIn an Antarctic penguin colony, 200 penguins are tagged and released. A year later, 100 penguins are captured and 4 of them are tagged. Students determine how many penguins are in the colony.How Many Yellow?Given the total number of balls in a bag and the probability for two colors, students find the number of balls for the third color.How Many Ways to Line Up?Students decide how many different ways they five students can order themselves as they line up for class.Gumballs There are some white gumballs and red gumballs left in a machine. Students find the probability of getting at least one red gumball.New FamilyA married couple wants to have four children. Students find the probability that at least one child will be a girl.Nickel and DimeStudents find the probability for different outcomes when tossing two coins.Four More FlipsStudents determine how many more tails are likely if a coin has already landed on tails twice.Bubble GumThe letters G, U, or M are printed inside bubble gum wrappers in a ratio of 3:2:1. Students use a simulation to find out how much bubble gum to buy to get a 3:2:1 ratio.A Large FamilyIf a family wants to have six children, what is the probability that there will be three boys and three girls? Students use a simulation to model the probability.No TelephoneUsing census data from 1960 and 1990 in two box plots, students compare the percentages of families that had phones.Pulse RateStudents compare two data sets of different sizes: one for students and one for athletes.Golf ScoresStudents are given two sets of golf scores for Rosa and Chen. They are asked to decide who is the better golfer by constructing and comparing box plots.How Much Taller?Given two sets of data about heights, students determine how much taller one group is than the other.Coin Jar Students determine the contents of a coin jar by sampling.Project Work TimeStudents can choose to work on and complete their project or get help if needed.

Students continue to extend their understanding of compound events by comparing independent and dependent events. This includes drawing the sample space to understand how the first event does or does not affect the second event. Students will solve problems with dependent compound events.Key ConceptsStudents will learn about the differences between dependent and independent events.Events are independent if the outcome of an event does not influence the outcome of the others.Events are dependent if the outcome of an event does influence the outcome of the others.The difference can be observed by drawing a diagram to represent the sample space. For dependent events, the sample space is smaller.Goals and Learning ObjectivesUnderstand the difference between independent and dependent compound events.Draw diagrams for dependent compound events.Solve compound event problems.

Students will begin to think about probability by considering how likely it is that their house will be struck by lightning. They will consider the relative likelihood of familiar events (e.g., outdoor temperature, test scores) on the continuum between impossible and certain. Students will discuss where on the continuum "likely," "unlikely," and "equally likely as unlikely" are.Key ConceptsAs students begin their study of probability, they look at the likelihood of events. Students have an intuitive sense of likelihood, even if no numbers or ratios are attached to the events. For example, there is clearly a better chance that a specific student will be chosen at random from a class than from the entire school.Goals and Learning ObjectivesThink about the concept of likelihood.Understand that probability is a measure of likelihood.Informally estimate the likelihood of certain events.Begin to think about why one event is more likely than another.SWD: Students with disabilities may need additional support seeing the relationships among problems and strategies. Throughout this unit, keep anchor charts available and visible to assist them in making connections and working toward mastery. Provide explicit think alouds comparing strategies and making connections. In addition, ask probing questions to get students to articulate how a peer solved the problem or how one strategy or visual representation is connected or related to another.

This MathWorld page gives information about permutations. Its real merit lies in the fact that there are examples for a clear understanding at various parts of the site and many links to learn detailed information about various aspects of permutations.