Understanding Matching vs. Coin Flipping Problems and Model Setup

Differentiating Between Matching and Coin Flipping Problems

Matching Problems Defined

• Matching problems inherently involve pairing things up.
• Key characteristic: Everything comes in pairs, and once a pairing is made, the items involved are typically removed from the pool or their availability changes.
• Dependence: Events are not independent. What is picked next depends on what was picked previously, as probabilities change based on prior selections.
• Example (Exam): Drawing lines from a list of words to a list of definitions.
  • One word goes with one definition.
  • Once a definition is used for a word, it cannot be used for any other word.
  • This means the probabilities for subsequent pairings change and depend on earlier choices.
• Example (Cell Phone Handout - $65$ phones total):
  • Imagine $65$ cell phones are collected and then randomly handed back to $65$ people.
  • First person: The probability of getting their own phone is $\frac{1}{65}$.
  • Second person: The probability of getting their own phone is not necessarily $\frac{1}{64}$.
    • The probability might be $\frac{1}{64}$ if the first person did not receive the second person's phone.
    • The probability could be $0$ (zero) if the first person did receive the second person's phone (meaning the second person's phone is no longer available in the bag).
  • This demonstrates dependence: what happens to the second person depends on what happened to the first person. Probabilities are constantly changing.
  • This is a classic matching problem because we are trying to pair up specific phones with their owners.

Coin Flipping Problems (Independent Events with Fixed Probabilities and Two Outcomes)

• These problems are characterized by three essential properties:
  1. Independent Events: The outcome of one event does not affect the outcome or probability of any other event.
  2. Fixed Probability: The probability of a specific outcome (success) remains constant for each event.
  3. Two Possible Outcomes: Each event has only two outcomes, typically labeled "success" and "failure" (or "heads" and "tails").
• Example (Verizon Cell Phone Provider Problem):
  1. What constitutes an independent event?
     • An event is one person saying who their cell phone provider is.
     • It is not "the class" or "collecting data." Data from $63$ people implies $63$ events.
     • Independence check: If one person has Verizon, does it affect the probability that another person has Verizon? No. Your choice does not influence anyone else's choice.
  2. Are these events independent? Yes, as established above. Your answer doesn't affect anyone else's answer.
  3. Do we have a fixed probability for each student?
     • According to the model (e.g., a Verizon rep's claim), the probability of any given student having Verizon is fixed (e.g., $40\%$ or $0.40$) for every student.
     • The probability does not change from one student to the next.
  4. Are there only two outcomes?
     • While there might be many cell phone providers (e.g., Verizon, AT&T, T-Mobile, Mint, etc.), for a coin flipping problem, we can categorize outcomes into two groups: Verizon customer or not a Verizon customer.
     • This effectively creates a binary outcome (two choices).
• Conclusion: Since the Verizon problem meets all three criteria (independent events, fixed probability, two outcomes), it is a coin flipping problem.

Setting Up the Coin Flipping Model (Using an App)

• The process involves a series of specific questions to configure the simulation.
• Question 1: What constitutes an independent event, and how many do we have?
  • Answer: One person's response about their cell phone provider.
  • Number of events: If data was collected from $63$ people, then we have $63$ events.
  • App configuration: Set the number of "coins" to $63$.
• Question 2: Which event do we care about? What counts as "success" (Heads)?
  • Answer: Being a Verizon customer.
  • Definition: A person saying they have Verizon is "heads"; saying they have anything else is "tails."
• Question 3: According to the model, what is the probability of heads?
  • CRITICAL POINT: This is the most common pitfall for students.
  • Emphasis: Always use the probability stated by the model or claim being tested, not the observed data from your collection (e.g., if you collected $41.3\%$ Verizon customers, do not use $41.3\%$).
  • Example: If the model claims $40\%$ of all IU students are Verizon customers, then the probability of heads is $0.40$. This is the claim we are testing; we are simulating the model, not our data or desired outcome.
  • App configuration: Set the probability of heads to $0.40$.

Running and Interpreting One Run of the Model

• First step: Always click "Run Once" when starting the simulation.
• What "Run Once" does: It flips $63$ (or whatever the set number) coins, simulating asking $63$ students about their cell phone provider under the assumption that the model (e.g., $40\%$ Verizon) is true.
• It then counts the number of "heads" (e.g., the number of Verizon customers in that simulated group of $63$).
• Result: One "run once" will produce a single number (e.g., $25$ Verizon customers out of $63$).
• Interpreting "What does one run of the model represent?" (A crucial, recurring question worth three points):
  1. It represents one instance of running the experiment.
  2. It does so assuming the model is correct (e.g., assuming $40\%$ of students are Verizon customers).
  3. It produces one possible outcome (e.g., $25$ Verizon customers) that you might expect to get from that single instance if the model were true. This highlights the "noisy" nature of the world; identical experiments will yield slightly different results due to randomness.