Welcome back to Professor Baker's Math Class! For Spring 2025, we are tackling Chapter 6: Probability, Randomness, and Uncertainty. This chapter bridges the gap between descriptive statistics and inferential statistics by teaching us how to quantify uncertainty. Below is a breakdown of the key concepts from our class notes to help you master these topics.

Part 1: The Foundations of Probability (6.1 & 6.2)

We begin by defining a Random Experiment, an activity where the outcome is uncertain, but the distinct possibilities are known. The set of all distinct outcomes is called the Sample Space, denoted by $S$.

Key Probability Laws:

  • Probabilities must always be between 0 and 1: $0 \le P(A) \le 1$.
  • The sum of probabilities for all outcomes in a sample space must equal 1.
  • The Complement Rule: The probability of an event not occurring is $P(A^c) = 1 - P(A)$.

We also explored Compound Events using Venn Diagrams:

  • Union ($A \cup B$): The event occurring in A or B (or both).
  • Intersection ($A \cap B$): The event occurring in A and B simultaneously.
  • Mutually Exclusive (Disjoint): Events that cannot happen at the same time, meaning $P(A \cap B) = 0$.

Make sure you memorize the General Addition Rule:

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

Part 2: Conditional Probability and Counting (6.3 & 6.4)

In the second half of the chapter, we move to more complex scenarios involving Conditional Probability. This is the probability of event A occurring given that event B has already occurred.

$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$

This leads us to the concept of Independence. Two events are independent if the occurrence of one does not affect the probability of the other. If events are independent, we can use the simplified multiplication rule: $P(A \cap B) = P(A) \cdot P(B)$.

The Art of Counting

Finally, we learned techniques to count outcomes without listing them all out, which is essential for calculating probabilities in large sample spaces.

  1. Fundamental Counting Principle: If you have $m$ ways to do one thing and $n$ ways to do another, there are $m \cdot n$ ways to do both.
  2. Permutations ($nPr$): Used when order matters (e.g., a license plate or a race finish).
  3. Combinations ($nCr$): Used when order does not matter (e.g., selecting a committee or a lottery hand).

The formula for combinations, which allows us to select $k$ items from $n$ distinct items, is:

$$_nC_k = \frac{n!}{k!(n-k)!}$$

Be sure to review the attached PDF notes for specific examples, such as the "Mississippi" distinguishable permutation problem and the lottery probability calculations. Good luck studying!