Curriculum / Math / 11th Grade / Unit 8: Probability and Statistical Inference / Lesson 2
Math
Unit 8
11th Grade
Lesson 2 of 13
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Lesson Notes
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Determine probabilities of events that are not mutually exclusive.
The core standards covered in this lesson
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.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.
The foundational standards covered in this lesson
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.
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.
The essential concepts students need to demonstrate or understand to achieve the lesson objective
Suggestions for teachers to help them teach this lesson
Venn diagrams are useful strategies for students to understand when events are mutually exclusive and when they are not. Each section of a two-circle Venn diagram (only, and, only, not) each would represent a separate branch on the tree diagram. It is not necessary for students to convert from one to the other, but it is important to choose a representation that makes sense and use it appropriately.
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
25-30 minutes
At a particular diner, everyone drinks coffee. However, some people drink their coffee with sugar, some with cream, and some with neither.
Describe the coffee preferences, described below, of the 30 people in the diner in the Venn diagram shown. Be sure to label the circles appropriately.
Which equation corresponds to which diagram? Can you write an equation for the third diagram?
Unions of Sample Sets is made available by New Visions for Public Schools under the CC BY-NC-SA 4.0 license. © 2017 New Visions for Public Schools. Accessed https://curriculum.newvisions.org/math/resources/resource/algebra-ii-unions-sample-sets/.
Below is a diagram that shows each member of the student council at a particular high school. Each member is represented by a labeled point. The students that are in circle $$J$$ are juniors, and the students who are in circle $$M$$ are male.
Use the diagram above to find the following probabilities:
Why does $$P(J\space\mathrm{or}\space M) \neq P(J) + P(M)$$?
Design a formula to calculate $$P(J\space\mathrm{or} \space M)$$ using any of the probabilities found above.
The Addition Rule, accessed on June 15, 2017, 8:42 a.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
5-10 minutes
At Mom’s diner, everyone drinks coffee. Let $${C=}$$ the event that a randomly selected customer puts cream in their coffee. Let $${S=}$$ the event that a randomly selected customer puts sugar in their coffee. Suppose that after years of collecting data, Mom has estimated the following probabilities:
$${P(C)=0.6}$$ $${P(S)=0.5}$$ $${P(C\space \mathrm{or}\space S)=0.7}$$
Estimate $${P(C\space\mathrm{and}\space S)}$$ and interpret this value in the context of the problem.
Coffee at Mom's Diner, accessed on June 15, 2017, 8:48 a.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.
The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.
Next
Calculate conditional probabilities.
Topic A: Probability
Determine probabilities of mutually exclusive events.
Standards
S.CP.A.1
S.CP.A.1S.CP.B.6S.CP.B.7
S.CP.A.3
Determine when events are independent and describe independent events using everday language.
S.CP.A.2S.CP.A.3S.CP.A.5
Calculate relative frequencies in two-way tables to analyze data and determine independence.
S.CP.A.4
Use conditional probability to make decisions about medical testing.
S.CP.A.2S.CP.A.3
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Topic B: The Normal Distribution
Describe the center, shape, and spread of distributions by reasoning visually about the mean, standard deviation, and shape of a histogram.
S.IC.A.1
Derive and calculate population percentages based on a normal distribution of data.
S.IC.A.2S.IC.B.4
Use $${z-}$$scores to identify population percentiles.
S.IC.B.4
Topic C: Statistical Inferences and Conclusions
Describe and compare statistical study methods.
S.IC.B.3S.IC.B.6
Use multiple random samples to estimate a population mean or proportion and verify the validity of the sampling method by analyzing the means and standard errors of samples.
Calculate and describe the margin of error in context and use larger sample sizes to minimize the margin of error.
Compare two treatments in experimental data and determine if the difference between the two treatments is significant.
S.IC.B.5
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