Welcome back, students! As we approach the exam for Chapters 4 and 5, it is time to solidify your understanding of probability distributions, counting rules, and the binomial theorem. The attached class notes provide the step-by-step solutions to the review packet, but let’s break down the core concepts you need to succeed.

1. Reading Probability Tables

A significant portion of Chapter 4 involves interpreting data from frequency and probability tables. Whether you are looking at Adjusted Gross Income (AGI) or living arrangements by age, remember these key rules:

  • Summation: The sum of all probabilities in a distribution must equal 1 (or 100%).
  • "Inclusive" Ranges: If a question asks for the probability of a value being between $X$ and $Y$, make sure to sum the probabilities of all categories that fall within that range.
  • Expected Value (Mean): To find the average of a discrete probability distribution, multiply each value $x$ by its probability $P(x)$ and sum the results: $$\mu = \sum [x \cdot P(x)]$$ (See the "Busy Phone Lines" problem in the notes for a worked example where $\mu = 2.817$).

2. Counting Principles: Order Matters?

Determining the total number of outcomes is crucial. You must decide if you are using the Fundamental Counting Principle, Permutations, or Combinations.

  • Fundamental Counting Principle: Used when choices are independent. For the telephone number problem in the notes, we calculated the total possibilities by multiplying the options for each digit ($9 \cdot 10 \cdot 10 \dots$).
  • Permutations ($nPr$): Use this when order matters. In the mutual funds example, you are selecting 4 funds for 4 specific quarters. Since the quarter matters, we use Permutations: $$_{30}P_4 = 657,720$$
  • Combinations ($nCr$): Use this when order does not matter.

3. Conditional Probability

When working with two-way tables (like the Age vs. Living Arrangement problem), pay close attention to the phrasing "given that". This changes your denominator.

For example, if asked for the probability a person is over 64 given they live with a spouse, you only look at the "With Spouse" column for your total, not the entire population.

4. The Binomial Probability Distribution

Chapter 5 focuses heavily on the Binomial Distribution. To use these formulas, you must have a fixed number of trials ($n$), two possible outcomes (success/failure), and a constant probability of success ($p$).

The Formula: $$P(X=x) = \binom{n}{x} p^x (1-p)^{n-x}$$ In the notes, we applied this to the "Pet Ownership" problem ($n=4, p=0.60$) to find probabilities of exactly 3, at least 3, or at most 3 households owning pets.

Mean and Standard Deviation: Don't forget the shortcuts for binomial distributions: $$\mu = np \quad \text{and} \quad \sigma = \sqrt{np(1-p)}$$

Study Tips

Review the attached PDF carefully. It contains the handwritten solutions for the blank test review. Pay special attention to how the word problems are translated into the mathematical notation (e.g., how "at most one" becomes $P(X \le 1)$).

Good luck with your studying! You have all the tools you need to ace this test.