Probability and Statistical Inference

Lesson 2

Math

Unit 8

11th Grade

Lesson 2 of 13

Objective


Determine probabilities of events that are not mutually exclusive.

Common Core Standards


Core Standards

  • 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.

Foundational Standards

  • 7.SP.C.5
  • 7.SP.C.6
  • 7.SP.C.7
  • 7.SP.C.8

Criteria for Success


  • Describe the meaning of events that are not mutually exclusive, such that the probability of $${P(A\space\mathrm{or}\space B)}$$ includes $${P(A\space\mathrm{and}\space B)}$$.
  • Use a tree diagram to describe the sample space of a chance experiment when events overlap and identify $$P(A),\space P(A \space\mathrm{or} \space B),\space P(\mathrm{not}\space A), {P(A\space\mathrm{and}\space B)}$$
  • Calculate the probability of an overlap event of $${P(A \space\mathrm{or}\space B)=P(A)+P(B)-P(A \space\mathrm{and}\space B)}$$ and describe that the rule is derived from the understanding that the probability of an “or” event should only count each outcome once and that subtracting the “and” ensures this. 
  • Describe complement events. 
  • Use Venn diagrams to represent probabilities that are not mutually exclusive.

Tips for Teachers


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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Anchor Problems


Problem 1

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. 

  • 10 people just like cream in their coffee.
  • 13 people like sugar in their coffee; 8 of these also like cream. 
  • 7 people like neither cream nor sugar in their coffee.

Guiding Questions

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Problem 2

Which equation corresponds to which diagram? Can you write an equation for the third diagram?

Guiding Questions

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References

New Visions for Public Schools Unions of Sample Sets

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/.

Modified by Fishtank Learning, Inc.

Problem 3

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: 

  • $$P(J)$$
  • $$P(M)$$
  • $$P(J\space\mathrm{or} \space M)$$
  • $$P(J\space\mathrm{and} \space M)$$

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. 

Guiding Questions

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References

Illustrative Mathematics The Addition Rule

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.

Modified by Fishtank Learning, Inc.

Target Task


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. 

References

Illustrative Mathematics Coffee at Mom's Diner

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.

Additional Practice


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.

  • Use the situations in Anchor Problem #1 and Anchor Problem #2 to also ask about compound probabilities. For example: 
    • What is the probability that you choose two random people in the diner, and both of them like cream and sugar in their coffee?
    • What is the probability that two students on the student council who are chosen on different weeks to make the announcements are female one week, and male and not junior on the second week?
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Lesson 1

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Lesson 3

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Probability

Topic B: The Normal Distribution

Topic C: Statistical Inferences and Conclusions

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