Introduction to Probability
Learning Objectives
- Define and explain probability terminology, likelihood, and experiments
- Use and, or, and not notation to describe events
- Use a tree diagram to list outcomes and compute probabilities
Why This Matters
Every time a weather app says "70% chance of rain," a doctor says "20% risk of side effects," or a video game shows "3% legendary drop rate," a probability is shaping your next decision. Probability isn't a math class artifact -- it's the formal language of uncertainty, and every field that deals with incomplete information runs on it. Learn the vocabulary and the rules here, and you'll understand what those numbers actually promise and what they don't.
How to Use This Simulation
- Choose an experiment (coin toss, die roll, card draw) and an event to track -- watch which outcomes in the sample space light up.
- Click Run 1, Run 10, Run 100, or Run 1,000 trials and watch the empirical probability converge toward the theoretical probability on the chart.
- Switch to the Double Coin Toss to see a tree diagram for compound experiments, and try events using and, or, not notation.
- Check the Explanation Panel below -- it updates as you interact and tells you why the numbers behave the way they do.
Sample Space
Tree Diagram
What's Happening
Quick Check
A weather app says there's a 30% chance of rain today. It doesn't rain. Your roommate says, "See? The app was wrong -- it said it would rain and it didn't." Which response best explains why the forecast wasn't necessarily wrong?
Try This
Select the "Board Game Die Roll" experiment. Before running any trials, write down the sample space (all six possible outcomes). Identify the outcomes in the event "greater than 3." Calculate the theoretical probability of this event by hand: favorable outcomes divided by total outcomes. Now run 100 trials in the simulation and compare your theoretical probability to the empirical result. Are they close? Should they be exactly equal after 100 rolls?
Select the "Double Coin Toss" experiment. The tree diagram shows four possible outcomes. (1) List all outcomes belonging to the event "exactly one head." (2) Calculate the theoretical probability of "exactly one head" using the tree diagram: count the favorable paths and divide by total paths. (3) Select "Exactly one head" from the event dropdown and run 500 trials. How close is the empirical probability to your calculation? (4) In one sentence, explain why the tree diagram is more reliable than gut instinct for counting outcomes of compound experiments.
A basketball player has made 7 free throws in a row during a game. The commentator says, "She's due for a miss -- nobody makes 8 in a row." Her season free-throw percentage is 78%. (1) State the theoretical probability she makes the next free throw. (2) Does her 7-shot streak change that probability? Explain why or why not. (3) The commentator is committing a well-known reasoning error. Name the error and explain in two sentences why the streak doesn't influence the next shot's probability.
Instructor Notes
Teaching Notes
This simulation is most effective when you let students run a small number of trials first (5-10 coin flips) and ask them to predict the next result before they flip. Many students will insist tails is "due" after a heads streak -- the convergence chart is your tool for dismantling that intuition visually. Let the chart do the arguing; don't lecture.
The Double Coin Toss tree diagram addresses a specific structural misconception: students consistently count "one head and one tail" as a single outcome rather than recognizing HT and TH as distinct paths. Have students predict how many outcomes give "exactly one head" before revealing the tree. Most will say 1 out of 3 rather than 2 out of 4.
Common Student Errors
- Treating HT and TH as the same outcome in compound experiments. The tree diagram makes their distinctness visible.
- Expecting empirical probability to match theoretical exactly after a small number of trials ("I flipped 10 times and got 7 heads, so the probability is 0.7"). This confuses a single sample with the long-run rate.
- The gambler's fallacy: believing outcomes are "due" after streaks. The simulation's trial-by-trial display shows that each trial is independent.
- Confusing "or" in everyday English (exclusive) with "or" in probability (inclusive). The die roll events make this visible: "even OR greater than 3" includes outcomes satisfying both conditions.
Discussion Questions
- If you flip a fair coin 100 times and get 60 heads, is the coin biased? What if you flip it 10,000 times and get 5,200 heads? How does the number of trials change your interpretation?
- A casino game has a 48% chance of winning each round. Why does the casino always make money in the long run, even though individual players sometimes win?
- Why does probability use inclusive "or" as the default? Can you think of a real-world situation where exclusive "or" is more natural?
Exam Connection
Typical exam questions ask students to list sample spaces, identify events as subsets of sample spaces, and compute theoretical probabilities using the equally-likely-outcomes formula. Tree diagram questions usually present a two-stage experiment and ask students to enumerate outcomes and compute P(event). The Double Coin Toss preset directly prepares students for this format. The and/or/not event selectors prepare students for notation they'll encounter in the addition and complement rules (Sims 11-12).