# Probability: The Basics --- ## Probability is the *likelihood* that something will happen. --- ## We express the probability that an event will occur as a number between 0 and 1. ---- <img src="https://i.imgur.com/B5ucJzB.png" style="width:800px;height:500px;"> *Source: mathisfun.com* --- # Definitions ## Chance Experiment: A process where we do something and record the results (which can vary). ---- # Chance Experiment **Examples:** * flipping a coin -- we record "heads or tails" * driving from Cullowhee to Franklin -- we record "time taken" * taking a Statistics course -- we record the grade ---- <h4> <span style="color:purple"> The possible outcomes outcomes of a chance experiment are called the sample space. </span> </h4> <img src="https://i.imgur.com/kCdeFGC.jpg" style="width:300px;height:150px;"> *Example:* Flipping a coin. The sample space is {heads, tails}. <h6> Photo by <a href="https://unsplash.com/@yakimadesign?utm_source=unsplash&utm_medium=referral&utm_content=creditCopyText">Jordan Rowland</a> on <a href="https://unsplash.com/s/photos/flipping-a-coin?utm_source=unsplash&utm_medium=referral&utm_content=creditCopyText">Unsplash</a> </h6> --- *Example:* Getting dressed. Bob only owns three shirts: red, green, and blue and two pairs of pants: black or brown. What is the *sample space* of the *chance experiment* of Bob getting dressed? ---- # Tree Diagrams ---- **Answer:** {(red,black), (red, brown), (green, black), (green,brown), (blue, black), (blue,brown)} --- # Events **Definition:** An **event** is a collection of outcomes. **Definition:** A **simple event** is one consisting of a single outcome. ---- ## Example We roll a fair 6-sided die and record the number that comes up. (Fair means each number is equally likely.) The *event* **"Roll a number less than or equal to 4"** consists of 4 outcomes. "Roll a number less than or equal to 4" = {1,2,3,4} --- ## How Probability Works 1. We *assign* a probability to each outcome (based on theory or data). 2. We use mathematical rules to work with these probabilities. --- ## Equally likely outcomes? For many experiments, we assume that each outcome has an **equally likely outcome** -- in this case the probability of each outcome is &NewLine; &NewLine; $$\frac{1}{\text{total number of outcomes}}$$ --- # Definition For an event $E$ in a sample experiment, the probability of $E$ is denoted $P(E)$ and is defined as &NewLine; &NewLine; $$P(E) = \frac{\text{number of outcomes in E}}{\text{total number of outcomes}} $$ --- ## Example **Experiment**: A coin is flipped three times and the results of each flip are recorded. What is the sample space for this experiment? ---- **Experiment**: Recording three coin flips. **Q1:** What is the sample space for this experiment? **Answer:** $\{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT \}$ ---- **Q2:** If we assume **equally likely outcomes**, what is the probability of each outcome? **Answer:** $\frac{1}{8}$ **Sample Space:** $\{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT \}$ has a size of 8. ---- **Q3:** What is the probability of getting exactly two heads?  **Event** "exactly 2 heads" (find the outcomes) **Answer:** $\frac{3}{8}$ ---- ---- What is the probability of getting at least two heads?  **Event** "at least 2 heads" (find the outcomes) ----