Chapter 6 - Part 1 (Discrete Probability Distribution)

Chapter Six: Discrete Probability Models

Introduction to Random Variables

• Focus on discrete random variables in probability models.

• A random variable (RV) is a symbol (usually X) representing outcomes of a scenario.

Definition and Examples

• Recall from Chapter Five: Example of sampling without replacement to represent blue skills with RV X.

• RV X can take values of 0, 1, or 2.

Discrete vs. Continuous Random Variables

Discrete Random Variables

• A discrete variable has a finite, limited set of possibilities.

• Countable outcomes, e.g., number of patients in a hospital (12, 13, 14, ...) has fixed possibilities.

• Example: In the interval from 13 to 16, possible values are 13, 14, 15, 16—totaling four possibilities.

Continuous Random Variables

• Continuous variables allow for an infinite set of outcomes.

• Example: Height of a student (can have decimals like 69.25, etc.), meaning outcomes cannot be listed comprehensively.

• Other examples: Length of a house, income amount, GPA.

Key Distinction

• Discrete measurements (e.g., number of students) can be counted exactly.

• Continuous measurements (e.g., weight) cannot be counted exactly due to infinite possibilities.

Properties of Discrete Probability Models

• Chapter Six will cover how to visualize and describe models.

• Must check three properties to define a valid probability model:

  1. Numerical Values: First row (outcomes) must have numerical values.

  2. Valid Probabilities: P(X) must range between 0 and 1.

  3. Total Probability: The sum of all probabilities must equal 1.

Example 1: Probability Distribution Table

• Scenario: Bag contains 3 red marbles and 8 blue marbles.

• Define RV X = number of red marbles selected in 2 picks without replacement.

Possible Outcomes of X

1. X = 0:

   • Probability none red: P(X=0) = (\frac{8}{11} \times \frac{7}{10} \approx 0.5091)

2. X = 1:

   • To find this value, recognize two scenarios exist (one red and one blue ).

   • Use complementary probability: P(X=1) = 1 - (P(X=0) + P(X=2)).

3. X = 2:

   • Probability both red: P(X=2) = (\frac{3}{11} \times \frac{2}{10} \approx 0.0545)

Final Probability Distribution Table

• P(X=0) = 0.5091

• P(X=1) = 0.4364

• P(X=2) = 0.0545

• Confirm: 0.5091 + 0.4364 + 0.0545 = 1.