Section 15.4: Compound Events

Property

A sample space is the list of all possible outcomes for a probability experiment. An event is a subset of the sample space.
Theoretical probability is calculated as:

Examples

• When rolling a standard six-sided die, the sample space is $\{1, 2, 3, 4, 5, 6\}$. The probability of the event 'rolling an even number' is $P(\text{even}) = \frac{3}{6} = \frac{1}{2}$.
• When flipping a coin, the sample space is $\{\text{heads}, \text{tails}\}$. This is a uniform model where $P(\text{heads}) = P(\text{tails}) = \frac{1}{2}$.
• In a class of 20 students, if a student is selected at random, the probability that any specific student like Sam is selected is $P(\text{Sam}) = \frac{1}{20}$.

Explanation

Understanding sample spaces and theoretical probability provides the foundation for analyzing more complex situations. By identifying all possible outcomes and counting favorable ones, we can calculate exact probabilities for events. This theoretical approach will be essential when working with compound events involving multiple steps or conditions.

Property

A tree diagram systematically displays all possible outcomes of compound events by creating branches for each outcome of the first event, then extending branches for each outcome of subsequent events from every existing branch.

Examples

Property

If an event A can occur in $m$ ways and event B can occur in $n$ ways, then events A and B can occur in $m \cdot n$ ways.
The Fundamental Counting Principle can be generalized to more than two events occurring in succession.

Examples

• If you have 5 shirts and 3 pairs of pants, you can create $5 \times 3 = 15$ different outfits.
• A restaurant offers 4 appetizers, 6 main courses, and 3 desserts. The total number of different three-course meals is $4 \times 6 \times 3 = 72$.
• To create a code, you choose a letter from the alphabet (26 choices) and then a single digit from 0-9 (10 choices). There are $26 \times 10 = 260$ possible codes.

Explanation

Instead of listing every single possibility, just multiply the number of choices for each step. It is a powerful shortcut to find the total number of combinations, like for outfits, meal choices, or passwords.