Math / 10th Grade / Unit 8: Probability
Students formalize their understanding of compound probability, develop an understanding of conditional probability, and understand and calculate permutations and combinations.
Math
Unit 8
10th Grade
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In Unit 8, Probability, students extend their understanding of probability from seventh grade to a more formal approach to probability by applying formulas and definitions.
Students begin the unit using visual representations of lists, tree diagrams, and Venn diagrams to find the probability of events that intersect or represent the union or complement of outcomes. Students also formalize their understanding of compound probability. Then, students develop a conceptual and procedural understanding of conditional probability and how this can be used to determine whether variables are independent. In application, students use this knowledge to solve a set of applications on medical testing. Finally, as an optional part of the unit (covering plus (+) Common Core State Standards), students understand and calculate permutations and combinations. The students develop an understanding that whether order matters or not can affect the sample space of the total combinations or the number of chosen items within a set of items.
In Algebra 2, students will continue their study of probability by studying statistical inference and making decisions using probability.
Pacing: 12 instructional days (10 lessons, 1 flex day, 1 assessment day)
This assessment accompanies Unit 8 and should be given on the suggested assessment day or after completing the unit.
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Internalization of Standards via the Unit Assessment
Internalization of Trajectory of Unit
The central mathematical concepts that students will come to understand in this unit
Terms and notation that students learn or use in the unit
The materials, representations, and tools teachers and students will need for this unit
Topic A: Conditional Probability and the Rules of Probability
Describe the sample space of an experiment or situation. Use probability notation to identify the “and,” “or,” and complement outcomes from a given sample space.
7.SP.C.8 S.CP.A.1
Determine the probability of events with replacement using tree diagrams, addition rules for mutually exclusive events, and multiplication rules for compound events.
7.SP.C.7 S.CP.A.2 S.CP.A.4 S.CP.B.6 S.CP.B.7
Determine the probability of events without replacement using tree diagrams, addition rules for mutually exclusive events, and multiplication rules for compound events.
7.SP.C.7 S.CP.A.2 S.CP.B.6 S.CP.B.7 S.CP.B.8
Determine the probability of events that are not mutually exclusive to formalize the addition rule.
S.CP.A.1 S.CP.A.2 S.CP.B.7
Describe conditional probability and develop the rule $$P(B|A)=\frac{P(A \space \mathrm{and} \space B)}{P(A)}$$.
S.CP.A.3
Determine whether events are independent.
S.CP.A.2 S.CP.A.3 S.CP.A.5
Calculate and analyze relative frequencies in two-way tables to make statements about the data and determine independence.
S.CP.A.4 S.CP.A.5 S.ID.B.5
Make decisions about medical testing based on conditional probabilities.
S.CP.A.3 S.CP.A.4 S.CP.A.5
Describe and apply the counting principle and permutations to contextual and non-contextual situations.
S.CP.B.9
Describe and apply the counting principle and combinations to contextual and non-contextual situations.
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The content standards covered in this unit
S.CP.A.1 — Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").
S.CP.A.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.
S.CP.A.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.
S.CP.A.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.
S.CP.A.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.
S.CP.B.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.
S.CP.B.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.
S.CP.B.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.
S.CP.B.9 — Use permutations and combinations to compute probabilities of compound events and solve problems.
S.ID.B.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.
7.SP.C.7 — Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
7.SP.C.8 — Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
Standards covered in previous units or grades that are important background for the current unit
7.SP.C.5 — Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
7.SP.C.6 — Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
8.SP.A.4 — Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?
Standards in future grades or units that connect to the content in this unit
S.IC.A.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 5 tails in a row cause you to question the model?
S.IC.B.5 — Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
S.IC.B.6 — Evaluate reports based on data.
S.MD.A.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.
S.MD.A.2 — Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
S.MD.A.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.
S.MD.A.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?
S.MD.B.7 — Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
S.MD.B.5 — Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
S.MD.B.6 — Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP6 — Attend to precision.
CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.
Unit 7
Circles