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### CR Conditional Probability and Rules of Probability

- SPCR.1 Describe events as subsets of a sample space and
- a Use Venn diagrams to represent intersections, unions, and complements.
- b Relate intersections, unions, and complements to the words and, or, and not.
- c Represent sample spaces for compound events using Venn diagrams.

- SPCR.2 Use the multiplication rule to calculate probabilities for independent and dependent events. 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.
- SPCR.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.
- SPCR.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.
- SPCR.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
- SPCR.6 Calculate the conditional probability of an event A given event B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.
- SPCR.7 Apply the Addition Rule and the Multiplication Rule to determine probabilities, including conditional probabilities, and interpret the results in terms of the probability model.
- SPCR.8 Use permutations and combinations to solve mathematical and real-world problems, including determining probabilities of compound events. Justify the results.

### MJ Making Inferences and Justifying Conclusions

- SPMJ.1 Understand statistics and sampling distributions as a process for making inferences about population parameters based on a random sample from that population.
- SPMJ.2 Distinguish between experimental and theoretical probabilities. Collect data on a chance event and use the relative frequency to estimate the theoretical probability of that event. Determine whether a given probability model is consistent with experimental results.
- SPMJ.3 Plan and conduct a survey to answer a statistical question. Recognize how the plan addresses sampling technique, randomization, measurement of experimental error and methods to reduce bias.
- SPMJ.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.
- SPMJ.5 Distinguish between experiments and observational studies. Determine which of two or more possible experimental designs will best answer a given research question and justify the choice based on statistical significance.
- SPMJ.6 Evaluate claims and conclusions in published reports or articles based on data by analyzing study design and the collection, analysis, and display of the data.

### ID Interpreting Data

- SPID.1 Select and create an appropriate display, including dot plots, histograms, and box plots, for data that includes only real numbers.
- SPID.2 Use statistics appropriate to the shape of the data distribution to compare center and spread of two or more different data sets that include all real numbers.
- SPID.3 Summarize and represent data from a single data set. Interpret differences in shape, center, and spread in the context of the data set, accounting for possible effects of extreme data points (outliers).
- SPID.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 to estimate areas under the normal curve.
- SPID.5 Analyze bivariate categorical data using two-way tables and identify possible associations between the two categories using marginal, joint, and conditional frequencies.
- SPID.6 Using technology, create scatterplots and analyze those plots to compare the fit of linear, quadratic, or exponential models to a given data set. Select the appropriate model, fit a function to the data set, and use the function to solve problems in the context of the data.
- SPID.7 Find linear models using median fit and regression methods to make predictions. Interpret the slope and intercept of a linear model in the context of the data.
- SPID.8 Compute using technology and interpret the correlation coefficient of a linear fit.
- SPID.9 Differentiate between correlation and causation when describing the relationship between two variables. Identify potential lurking variables which may explain an association between two variables.
- SPID.10 Create residual plots and analyze those plots to compare the fit of linear, quadratic, and exponential models to a given data set. Select the appropriate model and use it for interpolation.

### MD Using Probability to Make Decisions

- SPMD.1 Develop the probability distribution for a random variable defined for a sample space in which a theoretical probability can be calculated and graph the distribution.
- SPMD.2 Calculate the expected value of a random variable as the mean of its probability distribution. Find expected values by assigning probabilities to payoff values. Use expected values to evaluate and compare strategies in real-world scenarios.
- SPMD.3 Construct and compare theoretical and experimental probability distributions and use those distributions to find expected values.
- SPMD.4 Use probability to evaluate outcomes of decisions by finding expected values and determine if decisions are fair.
- SPMD.5 Use probability to evaluate outcomes of decisions. Use probabilities to make fair decisions.
- SPMD.6 Analyze decisions and strategies using probability concepts.