Introduction to Probability and the Monty Hall Paradox

Class Announcements and Curriculum Transition

Test Grading and Grades: * The teacher is nearly finished grading the recent tests. * Grades will be updated in the grade book by the end of the day. * Students will receive their physical tests back the following day. * The teacher plans to add bonus points to the scores because the test was intentionally designed to be "tricky."
Curriculum Shift from Trigonometry to Pre-Calculus and Probability: * The teacher and a colleague have decided to stop the current focus on trigonometry. * The rationale is that the remaining trigonometry material is primarily necessary for students planning to take Calculus. * Given that many students are already juniors and must take Pre-Calculus first, the focus will shift to Pre-Calculus concepts and Probability. * Probability was chosen because it is widely applicable in most other classes, including Pre-Calculus.

The Monty Hall Paradox Demonstration

Background of the Game Show: * The demonstration is based on a game show from the $1970s$ , $1980s$ , and running into the $1990s$ (referring to Let's Make a Deal). * The Scenario: There are three doors. Behind one door is a car (the desired prize), and behind the other two doors are goats (not the desired result).
The Game Mechanics: 1. A contestant chooses one of the three doors. 2. The host (represented by the teacher) reveals what is behind one of the other doors. 3. The host will always reveal a goat. 4. The contestant is then given the option to "stay" with their original choice or "switch" to the remaining closed door.
In-Class Game Trials: * Trial 1: The class voted to choose Door $3$ . The teacher revealed Door $2$ was a goat. The class decided to switch to Door $1$ . Result: The car was behind Door $1$ . The class switched and won. * Trial 2: The class chose Door $3$ as the initial choice. The teacher revealed Door $1$ contained a goat. The class was asked to stay or switch. One student wanted to stay; five wanted to switch. Some students eventually chose Door $3$ again. One trial resulted in a win when staying, another in a loss. * Trial 3: The class chose Door $1$ . The teacher checked the answer key and Door $3$ was revealed to be "coffee" (a mascot/stand-in for the goat). The class stayed and lost (Door $1$ and Door $3$ both had goats).
The Statistical Truth of the Paradox: * Although it appears to be a $50/50$ chance (one in two) between the last two doors, it is statistically more likely to win if you switch every single time.

Statistical Logic of the Monty Hall Problem

Initial Probabilities: * When first choosing a door, the probability of being correct is one out of three, or $\frac{1}{3}$ . * The probability that the car is behind the other two doors combined is $\frac{2}{3}$ .
New Information and Probability Distribution: * If you pick one door (e.g., Door $2$ ), that door has a $\frac{1}{3}$ chance of being right. * The remaining doors combined have a $\frac{2}{3}$ chance. * When the host reveals a goat behind one of those other doors, that $\frac{2}{3}$ probability is now concentrated entirely on the one remaining unpicked and unrevealed door. * Conclusion: The original choice still only has a $\frac{1}{3}$ chance of being correct, while the door you can switch to has a $\frac{2}{3}$ chance. Therefore, switching doubles the odds of winning.

Fundamental Probability Vocabulary

Probability Experiment: An action or a trial that has specific end results.
Outcome: The possible results of a probability experiment. * Example: When rolling a die, the possible outcomes are $1, 2, 3, 4, 5, ext{or } 6$ .
Event: A collection of one or more outcomes. * Example: Rolling an odd number on a die (the outcomes $1$ , $3$ , and $5$ constitute the event). * An event can also involve multiple actions, such as rolling a die twice.
Sample Space: The set of all possible outcomes for an experiment.

Determining Sample Spaces and Tree Diagrams

Calculating Total Outcomes: * To find the number of possible outcomes in a sample space, one must list all unique combinations of results.
Example: Flipping a Coin and Rolling a Die: * Action 1: Flip a coin. Possible outcomes are Heads ( $H$ ) and Tails ( $T$ ). * Action 2: Roll a six-sided die. Possible outcomes are $1, 2, 3, 4, 5, 6$ . * Tree Diagram Construction: 1. Start with two branches for the coin: $H$ and $T$ . 2. From the $H$ branch, create six sub-branches for the die numbers $1$ through $6$ . 3. From the $T$ branch, create six sub-branches for the die numbers $1$ through $6$ . * List of Potential Outcomes ( $12$ total): * $(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6)$ * $(T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)$ * Probability Calculation from Sample Space: * The probability of flipping a heads and rolling exactly a $4$ is $\frac{1}{12}$ , as it is only one specific outcome out of the $12$ possibilities.

Classroom Exercises and Practice Problems

Practice Problem 1: Flip Two Coins * Requirement: List all possible outcomes in the sample space using a tree diagram. * Step 1: The first coin results in Heads or Tails. * Step 2: From both the Heads and Tails branches of the first coin, create two branches (Heads and Tails) for the second coin.
Practice Problem 2: Flip Two Coins and Roll a Six-Sided Die * Requirement: List all possible outcomes in the sample space. * Step 1: Build upon the previous tree diagram of the two flipped coins. * Step 2: Add six branches to every final outcome of the two coins to represent the possible die rolls.
Final Instructions: The teacher notes that this unit moves away from the previous style of math. Students have five minutes to work on these exercises and are encouraged to practice now to ensure success on the upcoming test and the final exam.