Applications of Mathematics 12 -
Statistics and Probability (Chance and Uncertainty)

This sub-organizer contains the following sections:
Prescribed Learning Outcomes
Suggested Instructional Strategies
Suggested Assessment Strategies
Recommended Learning Resources

PRESCRIBED LEARNING OUTCOMES

It is expected that students will:

• use normal and binomial probability distributions to solve problems involving uncertainty.
• solve problems based on the counting of sets, using techniques such as the fundamental counting principle, permutations and combinations.
• model the probability of a compound event, and solve problems based on the combining of simpler probabilities.

It is expected that students will:

• find the population standard deviation of a data set or a probability distribution, using technology
• use z-scores and the normal distribution to solve problems
• use the normal approximation to the binomial distribution to solve problems involving probability calculations for large samples
• solve pathway problems, interpreting and applying any constraints
• use the fundamental counting principle to determine the number of different ways to perform multistep operations
• construct a sample space for two or three events
• classify events as independent or dependent
• solve problems, using the probabilities of mutually exclusive and complementary events

SUGGESTED INSTRUCTIONAL STRATEGIES

Decision-making situations that involve probability and uncertainty occur frequently in daily life. Students need opportunities to learn how to use the normal probability distribution as a tool when interpreting and solving such problems.

• In a large group discussion, point out that in problems involving probability, words often have a meaning different from their ordinary English usage. Help students distinguish between mathematical usage and common usage.
• Provide students with examples that do not fit the normal curve and ask them to determine why using the z-score is inappropriate, emphasizing that the z-score will not solve all probability problems. Some examples include:
  • a crown-and-anchor game in which every number is equally likely (i.e., no cluster about the mean)
  • the median age of teachers is 45 and their mean is 50 (i.e., not symmetric about the mean)
• Help students develop a list of important things to consider when calculating the probability of an event. The list could include:
  • trying to organize the events of the sample space using tables or tree diagrams when possible
  • recalling the difference between AND and OR and how it relates to union and intersection
  • determining if each part of the problem is a permutation or combination
  • trying to solve a simpler version of the problem by taking a small sample of the same event using the counting principle
• Use real-life problem situations to introduce new concepts. For example:
  • Monica and André play a tennis match in which the first person to win two games in a row (or a total of three games) wins the match. How many possibilities are there to a final outcome? (Using a tree diagram may be helpful.)
  • Is selecting three people from a class to be on the student council any different from picking a person from the class to be president, another to be treasurer, and a third to be secretary?
• Have students research the role of probability in the field of genetics (e.g., eye colour, blood type

SUGGESTED ASSESSMENT STRATEGIES

The study of statistics relies heavily on the theory of probability. Probability provides concepts and methods for dealing with uncertainty and for interpreting predictions based on uncertainty. Evidence of students’ abilities to use probability can best be collected in the context of experimental design and problem-solving activities.

Observe

• While students are working on problems involving uncertainty, look for evidence that they can:
  • use the appropriate terminology
  • distinguish between permutation and combination and provide an example of each
  • identify appropriate uses of the normal distribution
  • correctly calculate the mean and standard deviation of a data set or distribution
  • correctly use normal distribution and z-scores
  • make sound conclusions using data at hand
• To assess students’ work on statistics projects, look for evidence that they can:
  • compile appropriate data for the project
  • identify the problem to be analysed and explain how the data contribute to the solution
  • calculate appropriate statistics from the following: mean, standard deviation, and z-score
  • summarize data using tables, diagrams, or graphs
  • draw conclusions and inferences based on analyses of the data

Question

• To assess students’ understanding of basic concepts, pose questions such as the following:
  • What is the difference between possible outcomes and the probability of an outcome?
  • What is the difference in meaning between combination and permutation?
  • How does the use of AND and OR in the context of compound events affect the meaning?

RECOMMENDED LEARNING RESOURCES

Print Materials

• Applied Mathematics 12 available June 2001
• Exploring Advanced Algebra with the TI-83
• Exploring Statistics with the TI-82 Graphics Calculator
• Using the TI-81 Graphics Calculator to Explore Statistics

Software

• Secondary Math Lab Toolkit

© 2000 Copyright. All Rights Reserved. Curriculum Branch.
Maintained by: Mathematics Coordinator
Revised: November 29, 2000

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