New Colorado P-12 Academic Standards

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Content Area: Mathematics
Grade Level Expectations: High School
Standard: 3. Data Analysis, Statistics, and Probability

Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter)

Concepts and skills students master:

3. Probability models outcomes for situations in which there is inherent randomness

Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Understand independence and conditional probability and use them to interpret data. (CCSS: S-CP)
    • Describe events as subsets of a sample space5 using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events.6 (CCSS: S-CP.1)
    • Explain 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. (CCSS: S-CP.2)
    • Using the conditional probability of A given B as P(A and B)/P(B), interpret the 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. (CCSS: S-CP.3)
    • 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.7 (CCSS: S-CP.4)
    • Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.8 (CCSS: S-CP.5)
  2. Use the rules of probability to compute probabilities of compound events in a uniform probability model. (CCSS: S-CP)
    • Find the conditional probability of A given B as the fraction of Bs outcomes that also belong to A, and interpret the answer in terms of the model. (CCSS: S-CP.6)
    • 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. (CCSS: S-CP.7)
  3. Analyze the cost of insurance as a method to offset the risk of a situation. (PFL)

Inquiry Questions:

  1. Can probability be used to model all types of uncertain situations? For example, can the probability that the 50th president of the United States will be female be determined?
  2. How and why are simulations used to determine probability when the theoretical probability is unknown?
  3. How does probability relate to obtaining insurance? (PFL)

Relevance & Application:

  1. Comprehension of probability allows informed decision-making, such as whether the cost of insurance is less than the expected cost of illness, when the deductible on car insurance is optimal, whether gambling pays in the long run, or whether an extended warranty justifies the cost. (PFL)
  2. Probability is used in a wide variety of disciplines including physics, biology, engineering, finance, and law. For example, employment discrimination cases often present probability calculations to support a claim.

Nature Of:

  1. Some work in mathematics is much like a game. Mathematicians choose an interesting set of rules and then play according to those rules to see what can happen.
  2. Mathematicians explore randomness and chance through probability.
  3. Mathematicians construct viable arguments and critique the reasoning of others. (MP)
  4. Mathematicians model with mathematics. (MP)

5 the set of outcomes. (CCSS: S-CP.1)

6 "or," "and," "not". (CCSS: S-CP.1)

7 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. (CCSS: S-CP.4)

8 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. (CCSS: S-CP.5)

Content Area: Mathematics
Grade Level Expectations: Seventh Grade
Standard: 3. Data Analysis, Statistics, and Probability

Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter)

Concepts and skills students master:

2. Mathematical models are used to determine probability

Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Explain that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring.4 (CCSS: 7.SP.5)
  2. 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.5 (CCSS: 7.SP.6)
  3. Develop a probability model and use it to find probabilities of events. (CCSS: 7.SP.7)
    • Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. (CCSS: 7.SP.7)
    • Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.6 (CCSS: 7.SP.7a)
    • Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.7 (CCSS: 7.SP.7b)
  4. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. (CCSS: 7.SP.8)
    • Explain that the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. (CCSS: 7.SP.8a)
    • Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. (CCSS: 7.SP.8b)
    • For an event8 described in everyday language identify the outcomes in the sample space which compose the event. (CCSS: 7.SP.8b)
    • Design and use a simulation to generate frequencies for compound events.9 (CCSS: 7.SP.8c)

Inquiry Questions:

  1. Why is it important to consider all of the possible outcomes of an event?
  2. Is it possible to predict the future? How?
  3. What are situations in which probability cannot be used?

Relevance & Application:

  1. The ability to efficiently and accurately count outcomes allows systemic analysis of such situations as trying all possible combinations when you forgot the combination to your lock or deciding to find a different approach when there are too many combinations to try; or counting how many lottery tickets you would have to buy to play every possible combination of numbers.
  2. The knowledge of theoretical probability allows the development of winning strategies in games involving chance such as knowing if your hand is likely to be the best hand or is likely to improve in a game of cards.

Nature Of:

  1. Mathematicians approach problems systematically. When the number of possible outcomes is small, each outcome can be considered individually. When the number of outcomes is large, a mathematician will develop a strategy to consider the most important outcomes such as the most likely outcomes, or the most dangerous outcomes.
  2. Mathematicians construct viable arguments and critique the reasoning of others. (MP)
  3. Mathematicians model with mathematics. (MP)

4 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. (CCSS: 7.SP.5)

5 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. (CCSS: 7.SP.6)

6 For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected. (CCSS: 7.SP.7a)

7 For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies? (CCSS: 7.SP.7b)

8 e.g., "rolling double sixes" (CCSS: 7.SP.8b)

9 For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood? (CCSS: 7.SP.8c)