# Grades 9-12 Math Activity Set 2: Counting Techniques and More

This week’s lessons focus on probability, mutually exclusive and overlapping events, and recognizing independent and dependent events.

### Materials Needed

Each lesson has an answer sheet that provides guidance on the lesson and describes the key concept and questions from the lesson. Use this sheet to check your answers.

The student sheet can be printed and completed by the student, or the student can write their answers on a blank sheet of paper.

• Paper
• Pencil
• Number cube (you can use dice from a board game at home)
• Graphing calculator (if one is not available, random number generators are available online)

### Activity 1: Counting Techniques

One of the essential skills for success in probability is being able to count the number of possible outcomes of an experiment.

Use this lesson to learn useful counting techniques, including the counting principle and the formulas for permutations and combinations.

### Activity 2: Introduction to Probability

Probability is a measure of how likely an event is to occur.

Use this lesson to learn how to determine the theoretical and experimental probability of an event. Use a simulation to model a probability experiment and find the odds in favor of and against an event.

### Activity 3: Simulations and the Law of Large Numbers Lab

Estimate the probability that a randomly generated three-digit number contains at least two matching digits.

In this lab, you will calculate the theoretical probability and compare this to the experimental probability from the simulation.

If using an online random number  generator, choose to generate three random integers between 1 and 9.

### Activity 4: Mutually Exclusive and Overlapping Events

When two subsets of a sample space have no elements in common, the corresponding events are mutually exclusive events. When two subsets have at least one element in common, the corresponding events are overlapping events.

In this lesson, you will learn formulas for finding the probability of mutually exclusive events and overlapping events, and you will find probabilities by identifying complementary events.

### Activity 5: Independent and Dependent Events

Recognizing independent events and dependent events is one of the essential skills in probability theory. It is important to understand how independent and dependent events are related to conditional probability.

In this lesson, you will explore this connection through the use of probability formulas and a variety of real-world examples.