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# 2022 PA-D-2-3

last edited by 10 months ago

PA.D.2.3

PA.D.2.3  Define, compare, and contrast the probabilities of dependent and independent events.

## In a Nutshell

After calculating simple probabilities, students will explore compound probabilities that are either independent or dependent events. Students should understand the properties of dependent and independent events and be able to describe the differences between probabilities of these events.

## Teacher Actions

• Develop a Deep and Flexible Conceptual Understanding by knowing how and when to apply probability concepts to compare and contrast situations involving both simple and compound probabilities and developing the understanding that compound probability has more than 1 desired outcome or event.

• Develop Ability to Make Conjectures, Model and Generalize when describing how situations involving terminology like “and” , “or”,  “with replacement” , and “without replacement” affect the probability of an event occurring.

• Develop Mathematical Reasoning when calculating compound probability. Students should explore the ideas of dependence and independence by analyzing if and how the first desired event affects the sample space of the second desired event.
• Implement tasks that promote reasoning and problem-solving by giving students activities to calculate compound probability using words like “and”,  “or”,  “with replacement”,  and “without replacement”.  Students should look for similarities and differences between what each probability is asking them to find.

• Facilitate meaningful mathematical discourse by emphasizing the idea of dependence in compound probabilities by having students explain what happens to the sample space after the first event occurs in situations when a value is replaced compared to when a value is not replaced in a sample space.

• Pose purposeful questions when using  real-life examples for students to have a real-life context of dependent compound probability and guide students to explaining the reasoning for determining independent and dependent events as well as creating scenarios representing dependent events.

## Misconceptions

• Calculate and compare compound probability with and without replacement.

• An independent event occurs when the result of the second event is not affected by the result of the first event or an event with replacement.

• A dependent event occurs when the result of the second event is affected by the result of the first event or an event without replacement.

• Identify independent and dependent events in both mathematical situations (dice, cards) and in real-world situations (the amount of time worked and the amount paid).

• Students may confuse the properties of an independent event for a dependent event. An example of this may be that students think that the amount of time spent studying depends on their test score instead of their test score depending on the amount of time studying.

• Dependent events that use “without replacement” can be difficult for some students to determine an accurate probability.

## Prior Knowledge

• Represent possible outcomes using a probability continuum from impossible to certain (6.D.2.1).

• Determine the sample space for a given experiment and determine which members of the sample space are related to certain events. Sample space may be determined by the use of tree diagrams, tables or pictorial representations (6.D.2.2).

• Determine the theoretical probability of an event using the ratio between the size of an event and the size of the sample space (7.D.2.1).

• Calculate experimental probabilities and represent them as percents, fractions, and decimals between 0 and 1 (PA.D.2.1).

• Given a Venn diagram, determine the probability of the union of events, and intersection of events, and the complement of an event.  Understand the relationships between these concepts and the words “AND”, “OR”, and “NOT” (A1.D.2.2).

• Apply probability concepts to real-world situations to make informed decisions (A1.D.2.4).

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