Curriculum / Math / 10th Grade / Unit 8: Probability / Lesson 4
Math
Unit 8
10th Grade
Lesson 4 of 10
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Determine the probability of events that are not mutually exclusive to formalize the addition rule.
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.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.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.
The essential concepts students need to demonstrate or understand to achieve the lesson objective
Suggestions for teachers to help them teach this lesson
This lesson introduces Venn diagrams to visualize probabilities. 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) represents a separate branch on a tree diagram. It is not necessary for students to convert from one to the other, but important for students to be able to choose a visual 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
At a particular diner, everyone drinks coffee. However, some people drink their coffee with sugar, some with cream, some with both, and some with neither.
Describe the coffee preferences of the 30 people in the diner in the Venn diagram below. Be sure to label the circles appropriately.
Below is a diagram that shows all of the members 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 in circle “$$M$$” are male.
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
At Mom’s diner, everyone drinks coffee. Let $$C$$ be the event that a randomly selected customer puts cream in their coffee. Let $$S$$ be 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:
$$P(C)=0.6$$ $$P(S)=0.5$$ $$P(C \space or \space S)=0.7$$
Estimate $$ P(C \space and \space S)$$ 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.
Lesson 3
Lesson 5
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
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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