Section 5-1: Introduction to Probability

Welcome to Section 5-1, where we'll be exploring the fascinating world of probability! This section focuses on understanding and calculating probabilities in various scenarios. Get ready to think critically about how likely events are and how to quantify uncertainty.

Learning Objectives

  • Distinguish between different types of probability: We'll explore theoretical, empirical, and subjective probabilities.
  • Calculate mathematical probabilities: This includes understanding theoretical probability, distinguishing outcomes, probability of non-occurrence, probability of disjunction, and probability with area.

Key Concepts and Formulas

1. Basic Probability

The probability of an event is the ratio of favorable outcomes to the total number of possible outcomes, assuming all outcomes are equally likely. The formula is:

$$Probability = \frac{Number\ of\ Favorable\ Outcomes}{Total\ Number\ of\ Possible\ Outcomes}$$

Remember, a probability must be between 0 and 1, inclusive. A probability of 0 means the event can never occur, while a probability of 1 means the event will always occur.

2. Example: Flipping Coins

Let's say you flip two coins. What is the probability of getting two heads (HH)? There are four possible outcomes: HH, HT, TH, and TT. Only one outcome is HH, so the probability is:

$$P(HH) = \frac{1}{4} = 0.25$$

3. Probability of Non-Occurrence

The probability of an event *not* occurring is simply 1 minus the probability of the event occurring. If $P(A)$ is the probability of event A occurring, then the probability of A not occurring is:

$$P(not\ A) = 1 - P(A)$$

Example: Suppose the probability of selecting an English teacher you like is $\frac{7}{17}$. The probability of selecting a teacher you don't like is:

$$P(Teacher\ I\ don't\ like) = 1 - \frac{7}{17} = \frac{10}{17}$$

4. Probability of Disjunction (Or)

The probability of either event A or event B occurring (or both) is given by:

$$P(A\ or\ B) = P(A) + P(B) - P(A\ and\ B)$$

Where $P(A\ and\ B)$ is the probability of both A and B occurring.

Example: A librarian has 10 paperback algebra books, 15 paperback biology books, 21 hardbound algebra books, and 39 hardbound biology books. What is the probability that a book selected at random is an algebra book or a paperback book?

Let A be an algebra book and B be a paperback book. There are 85 books total. $P(A) = \frac{31}{85}$, $P(B) = \frac{25}{85}$, and $P(A\ and\ B) = \frac{10}{85}$.

$$P(A \text{ or } B) = \frac{31}{85} + \frac{25}{85} - \frac{10}{85} = \frac{46}{85} \approx 0.54 = 54\%$$

Empirical Probability

Empirical probability is based on observed data or experiments. It's the ratio of the number of times an event occurred to the total number of trials.

$$Probability = \frac{Number\ of\ times\ the\ event\ occurred}{Total\ number\ of\ trials}$$

Keep practicing, and you'll become a probability pro in no time!