Unit 8 - Probability and statistics
- Students should be able to communicate informed decisions by applying the Addition Rule to a problem involving the probability of compound events.
- The focus and emphasis should be on the understanding of the Addition Rule conceptually with limited emphasis on the manipulation of the equation.
- Students should have opportunities using various tools such as Venn Diagrams and two-way tables to help visualize events.
- Two-way tables can be used to reveal all the sample space. Venn diagrams can be used to show intersections of two or more variables.
- Students should be able to relate the conditional probability back to the conceptual interpretation of probability studied in previous courses.
- The focus and emphasis should be on the understanding of the Multiplication Rule conceptually with limited emphasis on the manipulation of the equation.
- Tree diagrams may be used to help students visualize events and probabilities of those events.
- Relevant questions should be answered based on the appropriate risk measures.
- Students should be able to explain how studies and/or models were used to determine the risk measures.
- Students should be able to recognize that the chances of a false positive or a false negative are not the same as the chances of having the condition or not having the condition given the test result.
- Students should be able to interpret and communicate the consequences, of making the false positive or false negative errors.
- Students should be able to interpret the notation for conditional probability in context.
- A false positive is the probability of a positive result given the condition is not present.
- A false negative is the probability of a negative result given the condition is present. Examples
- Given a positive test result, what are the chances the person has the illness measured in the screening test?
- Given that a person has the illness, what are the chances of them getting a positive test result on the screening test.
- Students should understand the terms permutation and combination and be able to solve simple problems involving selection and arrangements of objects in a line, including those involving repetition and restriction.
- The emphasis should be on the conceptual understanding and application of combinations and permutations.
- Students should be able to use and interpret formal notation to communicate about combinations and permutations (e.g.,
nPr and nCr to represent choosing r objects from n distinct objects). - A permutation is a special case of an arrangement.
- A combination is a special case of a selection.
- Repetition is a type of permutation where a repeat of elements from the set is allowed.
- Restriction is a type of permutation where each element is used only once, and a certain order is required.
- Students should be able to understand that the probabilities in a distribution are between 0 and 1, and that they should sum to 1.
- Students should define random variable and understand that the sample space consists of all the values the random variable can take.
- Through mathematically applicable explorations, students should develop an understanding that the expected value is the mean of the probability distribution.
- Students should be presented with culturally relevant problems where they are given the expected value and can interpret its meaning within context.
- Students should be able to calculate the probability of all possible outcomes of a given event and display the probability of each graphically.
- Students should understand that the sum of all the probabilities within one distribution will be 1 (100%).
- A chart showing every outcome and the resulting probabilities might be useful in graphing the probability distribution.
- Utilizing notation X as a discrete random variable denoting an outcome, P(X) is the probability the outcome occurs.
- Students should be able to find the probability of a certain quantity (e.g., P(X = 2)), and also the probability of a range of quantities (e.g., P(X > 2)).
- Students should be able to use the expected value of a random variable to make informed decisions.
- Students should consider net value or payoff when making decisions about real-life problems.
- Students should understand that two probability distributions can have the same expected value, but one may vary more than the other, and this should be considered in decision-making.
- It is not necessary to calculate the standard deviation of the probability distribution.
- Students should be able to identify, calculate, and interpret joint, marginal, and conditional relative frequencies in context of the data.
- Students should have opportunities to analyze meaningful, real-life data and recognize possible associations and trends in the data.
- Students should understand and apply concepts of sample space to describe categorical data.
- Respective symbolic notation: P(A and B) = P(A∩B) and P(A or B) = P(A∪B).
- Students should be able to use two-way frequency tables to find probabilities for unions and intersections.
- Students should have opportunities to use two-way frequency tables to compute conditional probabilities.
Textbook Connections
Module 12-2 Probability and Counting
Students solve problems involving using the rule for the probability of complementary events.
Module 12-4 Probability with Permutations and Combinations
Students solve problems involving probabilities of compound events using permutations and combinations.
Module 12-5 Probability and the Multiplication Rule
Students solve problems involving probability of independent and dependent events using the Multiplication Rule.
Module 12-6 Probability and the Addition Rule
Students solve problems involving events that are not mutually exclusive using the Addition Rule.
Module 12-7 Conditional Probability
Students solve problems involving conditional probability using the Multiplication Rule.
Module 12-8 Two-way Frequency Tables
Students decide if events are independent and approximate conditional probabilities using two-way frequency tables.
Module 12-2 Probability and Counting
Students solve problems involving using the rule for the probability of complementary events.
Module 12-4 Probability with Permutations and Combinations
Students solve problems involving probabilities of compound events using permutations and combinations.
Module 12-5 Probability and the Multiplication Rule
Students solve problems involving probability of independent and dependent events using the Multiplication Rule.
Module 12-6 Probability and the Addition Rule
Students solve problems involving events that are not mutually exclusive using the Addition Rule.
Module 12-7 Conditional Probability
Students solve problems involving conditional probability using the Multiplication Rule.
Module 12-8 Two-way Frequency Tables
Students decide if events are independent and approximate conditional probabilities using two-way frequency tables.