4. Independent and dependent events

Learning Goals:

By the end of this lesson, students will:

• identify what independent or dependent events are
• Differentiate between independent or dependent events
• Compare results between experiments undertaken
• Assigning probabilities to outcomes in the form of P(AB)
• Identify that for independent events, it can expressed in the form of P(A and B)=P(A)×P(B)P(A and B)=P(A)×P(B)

Syllabus Outcomes Addressed:

• Describe the results of two- and three-step chance experiments, with and without replacement, assign probabilities to outcomes, and determine probabilities of events; investigate the concept of independence(ACMSP246)
• distinguish informally between dependent and independent events

Syllabus Working Mathematically Indicators:

• explain the difference between dependent and independent events using appropriate examples (Communicating, Reasoning)
• explain the effect of knowing the result of the first step on the probability of events in two-step chance experiments, with and without replacement (Communicating, Reasoning)
• describe the validity of conditional statements used in descriptions of chance situations with reference to dependent and independent events, eg explain why if you toss a coin and obtain a head, then the probability of obtaining a head on the next toss remains the same (Communicating, Reasoning)

Teaching Strategies

• Use of concrete materials
• Literacy activities
• Student created problems
• Group learning
• Presentation of student work

Lesson Sequence

Resource:

• Coloured counters
• Dice
• Coins
• Spinners

This lesson's focus is to let students understand the difference between independent and dependent events in probability and be able to apply it to everyday situations.

Warm up (LIT):

As a class, the teacher will help students identify key words that will be used throughout the lesson before they start working in groups for their activity. The teacher will have a question up on the board, for example:

• Multiply the probabilities of each event to find the probability of both happening
• Bag with 10 marbles: 6 green, 3 blue, 1 purple. What's the probability of drawing a green marble, then a purple marble, if you replace the marble after the first draw

The teacher will then ask the class, what is some key phrases in this question that would help them recognize that this event is dependent. This warm will go for 5-10 minutes to allow students to ponder what makes an event independent or dependent. The teacher will do one example for the whole class and they will decide what the probability will be based on what the teacher's example is.

Activity 1:

Immediately after the warm up, students will then work in pairs and be given a few examples to work on and decide if those particular examples are independent or dependent. The teacher will work around the class ensuring the students are on tasks and ask a few students for their reasoning on why they chose events to be independent or dependent.

Activity 2

After the class have worked on the examples given, students will form groups of 3-4 and the teacher will supply each group with a multitude of manipulatives such as colour counters, dice, coins and spinners. Using those manipulatives students will make up at least 5 independent and 5 dependent events. Insuring that they include any key words that can helps other identify whether it is an independent or dependent event. Students are also tasked to create accompanying questions for their scenarios such as the ones seen in the warm up activity for the other groups to attempt and answer. After students have completed their scenarios and questions, they will hand their scenarios to the group next to them, and try to figure out which scenario is independent or dependent.

Homework:

Students are tasked on creating their own independent or dependent events using any sort of manipulatives they have at home. They can bring it to the following class to showcase what problems they have created.

Activities and image sourced from: https://www.teachingchannel.org/video/teaching-dependent-and-independent-events

Lesson Evaluation Question

• Were students able to create consistent independent or dependent events?
• Were the activities able to help students differentiate between independent and dependent events?
• Was there enough time allocated building students' knowledge during the warm up?
• Were all students equally participating in group work?