Lesson 1: 6.1 Two Types of Random Variables

What is the "test" to determine if a variable is discrete or continuous? (Can you count it on your fingers, or do you need a stopwatch/ruler?)

In a continuous world, why is the probability of a single, exact value (like exactly 6.000000 inches) always equal to zero?

Random variable

Discrete random variable

Continuous random variable

Probability distribution

Probability density curve

HSS-MD.A.1: Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

Understanding the nature of the variable helps students choose the right graphical representation (e.g., a histogram vs. a bar chart). It also prepares them for questions involving "area under the curve" for Normal distributions.

This section transitions students from the "data" we collected in Chapter 1 to the "theoretical models" of Chapter 6. It’s about understanding the "grammar" of the variables we analyze.

DOK 1: Categorize the following as discrete or continuous: The number of pets in a household, the weight of a gold bar, the number of heads in 10 coin flips.

DOK 2: Use a probability density curve (like a uniform rectangle) to find the probability that a random number falls between 0.3 and 0.7.

DOK 3: Explain why we use a table for discrete variables but an area-based graph (density curve) for continuous variables.

The Problem: Let $X$ be the number of cars a household owns. The distribution is:
0 cars: 0.09
1 car: 0.36
2 cars: 0.35
3 cars: 0.13
4 cars: 0.05
0.02 for 5 or more (let's assume exactly 5 for this exercise).

Task: Verify this is a valid discrete probability distribution. Then, find the probability that a randomly selected household owns at least 2 cars.

Decimals = Continuous: Students often think any number with a decimal is continuous. Point out that "The price of a gallon of gas" is discrete because it stops at thousandths of a dollar—there is a "next" possible value.

Support: "The Number Line Test." If you can pick up your pencil and jump from one possible value to the next, it's discrete. If you have to slide your pencil without lifting it to cover every value, it's continuous.

Extension: Introduce the Uniform Distribution. Have students calculate probabilities for a random number generator that picks any value between 0 and 1. Ask: "What is the height of the rectangle if the total area must be 1?"

Teacher assigns examples from the textbook and other resources.

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