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High School – Conditional Probability and the Rules of Probability Essential Questions: 1. How can we gather, organize and display data to communicate and justify results in the real world? 2. How can we analyze data to make inferences and/or predictions, based on surveys, experiments, probability and observational studies? Essential Vocabulary : independent events, conditional probability, two-way frequency table, independence, addition rule for probability, general multiplication rule for probability, permutation, combination HS.S-CP.2: Describe events as subsets of a sample space (the set of outcomes) using the characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events(“or,” “and,” “not”). Enduring Understandings Students will know… Students will understand… Students will be able to… 1. that events can be described as subsets of 1. use characteristics to describe events. sample space. HS.S-CP.2: Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. Enduring Understandings Students will be able to… Students will understand… Students will know… 1. that two events A and B are independent if 1. justify that two events are independent 1. independent events. through the use of probability rules. the probability of A and B occurring together is the product of their probabilities. HS.S-CP.3: Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. Enduring Understandings Students will be able to… Students will understand… Students will know… 1. determine the conditional probability of an 1. that the conditional probability of A given 1. conditional probability event given that another event has already B is the same as the probability of A, and occurred. the conditional probability of B given A is the same as the probability of B. HS.S-CP.4: Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. Enduring Understandings Students will be able to… Students will know… Students will understand… 1. construct a two-way frequency table from 1. two-way frequency tables 1. that frequency tables can be used to a set of data and calculate conditional determine conditional probabilities and probabilities. independence. HS.S-CP.5: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. Enduring Understandings Students will know… Students will understand… Students will be able to… 1. conditional probability and independence 1. that conditional probability and 1. recognize conditional probability and independence occur in many everyday independence in everyday situations. situations and will be able to discuss the role they play in these situations. HS.S-CP.6: Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. Enduring Understandings Students will understand… Students will be able to… Students will know… 1. that conditional probability can be 1. construct and appropriate model that 1. conditional probability. modeled in a variety of ways. displays the information needed to calculate the conditional probability. HS.S-CP.7: Apply the Addition Rule, P (A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. Enduring Understandings Students will know… Students will understand… Students will be able to… 1. Addition Rule for probability 1. that when using the Addition Rule for 1. compute the probability of A or B using probability that the probability of A and B the Addition Rule for probability. should not be counted twice. HS.S-CP.8: (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model. Enduring Understandings Students will be able to… Students will know… Students will understand… 1. General Multiplication Rule for probability 1. that when using the General Multiplication 1. compute the probability of A and B using the General Multiplication Rule. Rule that if events are dependent conditional probabilities used. HS S-CP 9: (+) Use permutations and combinations to compute probabilities of compound events and solve problems. Enduring Understandings Students will know… Students will understand… Students will be able to… 1. permutation and a combination 1. that there is a difference between a 1. compute probabilities of compound events permutation. using permutations and combinations High School – Making Inferences and Justifying Conclusions Essential Questions: 1. How can we gather, organize and display data to communicate and justify results in the real world? 2. How can we analyze data to make inferences and/or predictions, based on surveys, experiments, probability and observational studies? Essential Vocabulary : random sampling, statistic, parameter, population, sample, theoretical and experimental statistics, an experiment and an observational study, mean, proportion and margin of error, significance HS.S-IC.1: Understand statistics as a process for making inferences about population parameters based on a random sample from that population. Enduring Understandings Students will be able to… Students will understand… Students will know… 1. make inferences about a population from a 1. that statistics is a process for making 1. random sample random sample. inferences about population parameters 2. statistic based on a random sample from that 3. parameter population. 4. population 5. sample HS.S-IC.2: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5tails in a row cause you to question the model? Enduring Understandings Students will know… Students will understand… Students will be able to… 1. theoretical and experimental statistics 1. that consistent results with the given event 1. design a simulation that models a desired are important. event. HS.S-IC.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. Enduring Understandings Students will be able to… Students will know… Students will understand… 1. recognize a survey, an experiment and an 1. the components of a survey 1. that randomization relates to sample observational study. 2. an experiment and an observational study surveys, experiments, and observational 2. explain the differences between a survey, studies. experiment, and observational study. HS.S-IC.4: Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. Enduring Understandings Students will be able to… Students will understand… Students will know… 1. estimate a population mean or proportion 1. that repeated simulation is a way of 1. mean and develop a margin of error by the use of estimating a population mean or 2. proportion simulation. proportion and can be used to develop an 3. margin of error estimate of a margin of error. HS.S-IC.5: Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. Enduring Understandings Students will know… Students will understand… Students will be able to… 1. significance 1. that statistical significance is important. 1. decide if the differences between parameters are significant. HS.S-IC.6: Evaluate reports based on data. Enduring Understandings Students will understand… 1. that statistics are important in evaluating the options presented in a report Students will know… 1. how to read a report Students will be able to… 1. evaluate reports based on data. High School – Interpreting Categorical and Quantitative Data Essential Questions: 1. How can we gather, organize and display data to communicate and justify results in the real world? 2. How can we analyze data to make inferences and/or predictions, based on surveys, experiments, probability and observational studies? Essential Vocabulary: categorical data, center, shape and spread, extreme data points, joint, marginal, and conditional relative frequencies, slope, intercept, correlation coefficient, correlation, causation HS.S-ID.1: Represent data with plots on the real number line (dot plots, histograms, and box plots). GFPS will also address Categorical Data in this Benchmark High School Enduring Understandings Students will know… Students will understand… Students will be able to… 1. categorical data 1. that quantitative and categorical single 1. construct graphical representations of variable data can be represented categorical and quantitative data (dot plots, graphically. histograms, box plots, bar graphs, pie graphs, stem and leaf plots . . .) HS.S-ID.2: Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. High School Enduring Understandings Students will be able to… Students will understand… Students will know… 1. use the appropriate measure of center and 1. how to recognize the differences in shapes 1. that based on the shape of a distribution spread to describe a distribution. that different measure of center and spread of different distributions. are used to describe the distribution. HS.S-ID.3: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). High School Enduring Understandings Students will know… Students will understand… Students will be able to… 1. center 1. that data sets can be compared and 1. describe the center, shape and spread of a 2. shape contrasted using center, shape and spread. data set. 3. spread 2. identify any extreme data points. 4. extreme data points HS.S-ID.4: Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables, and Montana Indian data sources to estimate areas under the normal curve. Students will know… 1. the properties of a normal distribution Enduring Understandings Students will understand… 1. that mean and standard deviation of a data set can be used to estimate population percentages. Students will be able to… 1. use calculators, spreadsheets, and tables to estimate areas under the normal curve. 2. calculate mean and standard deviation HS.S-ID.5: Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. Enduring Understandings Students will understand… Students will be able to… Students will know… 1. that possible associations and trends may 1. construct a two way frequency table 1. joint occur in data. 2. calculate the various joint, marginal and 2. marginal conditional relative frequencies. 3. conditional relative frequencies HS.S-ID.6: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. b. Informally assess the fit of a function by plotting and analyzing residuals. c. Fit a linear function for a scatter plot that suggests a linear association. Enduring Understandings Students will be able to… Students will know… Students will understand… 1. represent two quantitative variables on a 1. that the appropriate selection of a scatter plot and calculate an appropriate regression model begins with a scatter plot regression model for the data (using and is then further analyzed by the use of appropriate technology). residuals. 2. that two quantitative variables can be displayed graphically on a scatter plot and there are various types of regression models that can used to model the data HS.S-ID.7: Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Enduring Understandings Students will know… Students will understand… Students will be able to… 1. slope (rate of change) and intercept 1. that slope as a rate of change in the context 1. given a linear model, identify the slope and of the data and intercept in the context of the intercept. the data needs to be interpreted. HS.S-ID.8: Compute (using technology) and interpret the correlation coefficient of a linear fit. Benchmark Enduring Understandings Students will know… Students will understand… Students will be able to… 1. correlation coefficient 1. that technology can compute the linear 1. use technology to compute the linear model and the correlation coefficient model and the correlation coefficient. HS.S-ID.9: Distinguish between correlation and causation. Enduring Understandings Students will know… Students will understand… 1. correlation and causation 1. that correlation does not imply causation. Students will be able to… 1. distinguish between correlation and causation. High School – Using Probability to Make Decisions Essential Questions: 1. How can we gather, organize and display data to communicate and justify results in the real world? 2. How can we analyze data to make inferences and/or predictions, based on surveys, experiments, probability and observational studies? Essential Vocabulary : random variable, expected value, theoretical probability, empirical probability, fair decision HS.S-MD.1: (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions. Enduring Understandings Students will be able to… Students will understand… Students will know… 1. assign probabilities to random variables 1. that a random variable for a quantity of 1. random variable and graph the results. interest can be defined by assigning a numerical value to each event in a sample space. HS.S-MD.2: (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. Enduring Understandings Students will be able to… Students will know… Students will understand… 1. expected value 1. that the expected value in a given situation 1. calculate expected values. can be interpreted as the mean of the probability distribution. HS.S-MD.3: (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes. Enduring Understandings Students will be able to… Students will understand… Students will know… 1. calculate the theoretical probabilities for a 1. that in a given situation the theoretical 1. theoretical probability given distribution. probabilities can be calculated to find the expected value of a certain outcome. HS.S-MD.4: (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households? Enduring Understandings Students will be able to… Students will know… Students will understand… 1. calculate the empirical probabilities for a 1. empirical probability 1. that in a given situation the empirical given distribution. probabilities can be calculated to find the expected value of a certain outcome. HS.S-MD.5: (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. a. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast food restaurant. b. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident. Enduring Understandings Students will know… Students will understand… Students will be able to… 1. expected value. 1. that there is a difference between the 1. calculate expected values of various events expected value and the payoff of various and make decisions based upon these events. values. HS.S-MD.6: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). Enduring Understandings Students will know… Students will understand… Students will be able to… 1. fair decision 1. that probability is used in decision making. 1. calculate the probability of an outcome of a given situation and determine if it is fair or not by the definition of fair for that event. HS.S-MD.7: (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). Enduring Understandings Students will know… Students will understand… Students will be able to… 1. probability’s role 1. that probability is used in decision making. 1. use probability to make a decision giving the best chance of a desired outcome.
